MYMathApps Index Page

\(1^\text{st}\)-moment of the mass

3 Multiple Integrals in Rectangular Coordinates: Review of Center of Mass of a Bar

(\(2\)-sided) limit

1 Intuitive Limits and Continuity: 2-Sided Limits and Continuity

(\(2\)-sided) limit

3 Limits and Continuity: Intuitive Limits for Functions of 1 Variable (Review)

(\(2\)-sided\)) limit is negative infinity

1 Intuitive Limits and Continuity: 2-Sided Limits and Continuity

(\(2\)-sided\)) limit is positive infinity

1 Intuitive Limits and Continuity: 2-Sided Limits and Continuity

(\(2\)-sided\)) limit is negative infinity

3 Limits and Continuity: When Limits Do Not Exist

(\(2\)-sided\)) limit is positive infinity

3 Limits and Continuity: When Limits Do Not Exist

2D Stokes' Theorem

3 Green's Theorem: Alternate Notations - 2D Stokes' Theorem

2D Gauss' Theorem

3 Green's Theorem: Alternate Notations - 2D Gauss' Theorem

4D Cylindrical Coordinates

3 Coordinate Systems: Cylindrical and Spherical Coordinates in Higher Dimensions (Optional)

4D Spherical Coordinates

3 Coordinate Systems: Cylindrical and Spherical Coordinates in Higher Dimensions (Optional)

4-volume

3 Multiple Integrals as Riemann Sums: Riemann Sum Definition of Triple Integrals

A Vector Potential is Unique up to a Curl-Free Field

3 Divergence, Curl and Potentials: Finding Vector Potentials

A Scalar Potential is Unique up to a Constant

3 Divergence, Curl and Potentials: Finding Scalar Potentials

Absolute Value of the Triple Product

3 Cross Product: Geometric Interpretation of Triple Products

absolute minimum

3 Functions of Several Variables: Functions of \(1\) Variable

3 Max-Min Problems: Functions of \(2\) Variables

absolute maximum

3 Functions of Several Variables: Functions of \(1\) Variable

3 Max-Min Problems: Functions of \(2\) Variables

absolutely convergent

2 Convergence of General Series: The Absolute Convergence Test

abstract form

3 Lines and Planes: Lines in \(\mathbb{R}^2\) - More on the Normal Equation

abstract form

1 Vectors: Parametric Lines

“abuse of notation”

3 Directional Derivatives and Gradients: Derivative Along a Curve and the Gradient

acceleration of gravity

2 Applications of Differential Equations: Falling near the Earth

Add a Multiple of One Row to Another Row (<strong>Row Operation III</strong>)*

3 Cross Product: Properties of Determinants

Addition and Subtraction of Power Series

2 Power Series: Addition and Subtraction

Additive

3 Cross Product: Properties of Determinants

alternating

2 Convergence of General Series: The Alternating Series Test

alternating harmonic series

2 Convergence of General Series: The Alternating Series Test

alternating series bound on the error in the approximation

2 Convergence of General Series: Approximating the Series - Alternating Series Bound

angle between the lines

3 Lines and Planes: Parallel or Intersecting Lines in \(\mathbb{R}^2\)

angle between the planes

3 Lines and Planes: Parallel and Intersecting Planes

angle \(\theta\) between the line and the plane

3 Lines and Planes: Parallel and Intersecting Lines and Planes in \(\mathbb{R}^3\)

angle

1 Vectors: Angles

angle

3 Dot Product: Angles

angle between the lines

1 Vectors: Intersections of Lines

angular velocity

3 Cross Product: Application - Balancing Torques

angular acceleration

2 Mass and Center of Mass: Moment of Inertia - Bar with Non-Uniform Density

angular acceleration

3 Cross Product: Application - Balancing Torques

antiderivative

2 Antiderivatives: Definition of Antiderivatives

antiderivative

3 Divergence, Curl and Potentials: Generalizing Antiderivatives

antiderivative

1 Antiderivatives: Definition of Antiderivatives

Antisymmetry (<strong>Row Operation I</strong>)*

3 Cross Product: Properties of Determinants

application point

3 Cross Product: Application - Balancing Torques

arbitrarily

2 Sequences: Precise Definiton of Limits

arbitrarily

1 Precise Limits and Continuity: Precise Limits at Infinity

Arc Length of any Curve

2 Arc Length and Surface Area: Differential of Arc Length

Arc Length of the Graph of a Pair of Functions in 3D

2 Arc Length and Surface Area: Arc Length - 3-Dimensional Space

arc length

3 Line Integrals: Arc Length

Arc Length of the Curve

3 Properties of Curves: Arc Length, Arc Length Parameter, and Speed

arc length function

3 Properties of Curves: Arc Length, Arc Length Parameter, and Speed

Arc Length of a Polar Curve

2 Polar Coordinates: Arc Length of a Polar Graph

Arc Length of a Parametric Curve

2 Arc Length and Surface Area: Arc Length of a Parametric Curve

Arc Length of a Parametric Curve in 3D

2 Arc Length and Surface Area: Arc Length - 3-Dimensional Space

Arc Length of the Graph of a Function

2 Arc Length and Surface Area: Arc Length of the Graph of a Function

arc length

2 Arc Length and Surface Area: Arc Length - 3-Dimensional Space

area under \(f(x)\) above the interval \([a,b]\)

1 Antiderivatives: Area as an Antiderivative

area

3 Multiple Integrals in Rectangular Coordinates: Applications of Double Integrals - Centroid of a Region

Area as a \(y\)-Integral

1 Applications of Integration: Area as a \(y\)-Integral

Area Between Two Curves

1 Applications of Integration: Area Between Two Curves

Area Below a Curve

1 Applications of Integration: Area Below a Curve

area function for \(f(x)\)

2 Antiderivatives: Area as Antiderivative

Area of a Triangle

3 Cross Product: Areas

Area of a Parallelogram

3 Cross Product: Areas

area function for \(f(x)\)

1 Antiderivatives: Area as an Antiderivative

Area of a Polar Rectangle

3 Multiple Integrals in Curvilinear Coordinates: Polar Coordinates - Grid Cell

Area under a Curve as an Integral

1 Riemann Sums: Riemann Sums and Areas

area under \(f(x)\) above the interval \([a,b]\)

2 Antiderivatives: Area as Antiderivative

area of a polar rectangle

3 Multiple Integrals in Curvilinear Coordinates: Polar Coordinates - Grid Cell

Area Enclosed by a Polar Graph

2 Polar Coordinates: Area Enclosed by a Polar Graph

Area Between Two Curves

2 Area and Average Value: Area Between Two Curves

Area Between Polar Graph

2 Polar Coordinates: Area Between Polar Graphs

Area Below a Curve

2 Area and Average Value: Area Below a Curve

Area as a \(y\)-Integral

2 Area and Average Value: Area as a \(y\)-Integral

Area under a Curve as an Integral

2 Riemann Sums: Riemann Sums and Areas

asymptotes

3 Curves and Surfaces: Quadratic Curves - Part 1

asymptotic lines

3 Curves and Surfaces: Quadratic Curves - Part 1

asymptotic cone

3 Curves and Surfaces: Quadratic Surfaces - Part 1

autonomous differential equations

2 Applications of Differential Equations: undefined

Average Value and Area

2 Area and Average Value: Average Value and Area

average value of a function

1 Applications of Integration: Average Value of a Function

average value of a function

2 Area and Average Value: Average Value of a Function

Average Value of a Function

2 Area and Average Value: Average Value of a Function

Average Value of a Function

1 Applications of Integration: Average Value of a Function

average radius of the cone

2 Arc Length and Surface Area: Area of a Surface of Revolution

Average Value and Area

1 Applications of Integration: Average Value and Area

average value

3 Line Integrals: Average Value

axes

3 Coordinate Systems: 2D Rectangular Coordinates

axis

3 Curves and Surfaces: Quadratic Surfaces - Part 1

azimuthal angle

3 Coordinate Systems: 3D Spherical Coordinates

Balancing problems

2 Applications of Differential Equations: undefined

Best Fit Parabola

2 Numerical Integration: Simpson's Rule

bounded

2 Sequences: Bounded Sequences

Bounded, Monotonic Sequence Theorem

2 Sequences: Bounded, Monotonic Sequence Theorem

bounded above

2 Sequences: Bounded Sequences

bounded below

2 Sequences: Bounded Sequences

By inspection

3 Divergence, Curl and Potentials: Finding Scalar Potentials

Cartesian Coordinate System

3 Coordinate Systems: 2D Rectangular Coordinates

case work

3 Max-Min Problems: Second Derivative Test - Examples

center

2 Power Series: What is a Power Series?

Center of Mass of a Bar with Non-Uniform Density

2 Mass and Center of Mass: Center of Mass - Bar with Non-Uniform Density

center of mass

2 Mass and Center of Mass: Center of Mass - Bar with Non-Uniform Density

center of mass

3 Line Integrals: Mass, Center of Mass and Centroid

3 Multiple Integrals in Rectangular Coordinates: Review of Center of Mass of a Bar

3 Parametric Surfaces and Surface Integrals: Applications of Double Integrals - Center of Mass

Center of Mass for Discrete Masses

2 Mass and Center of Mass: Center of Mass - Discrete Masses

centroid

3 Line Integrals: Mass, Center of Mass and Centroid

3 Multiple Integrals in Rectangular Coordinates: Applications of Double Integrals - Centroid of a Region

3 Parametric Surfaces and Surface Integrals: Centroid

Chain Rule for Functions with \(4\) Intermediate Variable

3 Chain Rule: Chain Rule with More Intermediate Variables

Chain Rule for Functions of \(1\) Variable

1 Chain Rule & Implicit Differentiation: Chain Rule

Chain Rule for Functions of \(1\) Variable

3 Chain Rule: Chain Rule with \(1\) Intermediate Variable

Chain Rule for Functions with \(2\) Intermediate Variables

3 Chain Rule: Chain Rule with \(2\) Intermediate Variables

Chain Rule for Functions with \(2\) Intermediate and \(2\) Independent Variables

3 Chain Rule: Chain Rule with More Independent Variables

Change Limits

2 Integration by Substitution: Definite Integrals - Changing Limits

change of variables

1 Integration by Substitution: Indefinite Integrals

change in the dependent variable

3 Linear Approximation: Differentials with \(3\) or more Variables

change of position

1 Vectors: Displacements

change of variables

2 Integration by Substitution: Indefinite Integrals

change of position

3 Vectors: The Vector Between Two Points - Displacement

changes in the independent variables

3 Linear Approximation: Differentials with \(3\) or more Variables

Changing Limits in Integration by Substitution

1 Integration by Substitution: Definite Integrals - Changing Limits

Changing Limits in Integration by Substitution

2 Integration by Substitution: Definite Integrals - Changing Limits

changing limits

2 Integration by Substitution: Definite Integrals - Changing Limits

changing limits

1 Integration by Substitution: Definite Integrals - Changing Limits

charge density

3 Divergence, Curl and Potentials: The Divergence of a Gradient (\(\vec\nabla\cdot\vec\nabla f\)) - The Laplacian

circle

3 Curves and Surfaces: Quadratic Curves - Part 1

circular cylinders

3 Curves and Surfaces: Quadratic Surfaces - Part 2

circulation of the fluid

3 Line Integrals: Flow and Circulation

circulation of the vector field

3 Line Integrals: Flow and Circulation

Clairaut's Theorem

3 Higher Order Derivatives: Clairaut's Theorem

coefficient of viscosity

2 Applications of Differential Equations: Falling with Air Resistance

coefficients

2 Power Series: What is a Power Series?

comparison series

2 Convergence of Positive Series: Overview

comparison bound on the error in the approximation

2 Convergence of Positive Series: Approximating the Series - Comparison Bounds

Comparison Test

2 Improper Integrals: Integrals Improper at a Finite Endpoint - Comparison Test

complete surface

3 Gauss' Theorem: The Theorem

complex conjugate

2 Partial Fractions: Complex Methods (Optional)

component or scalar projection of \(\vec v\) perpenducular to \(\vec u\)

1 Vectors: Scalar Projections

component or scalar projection of the vector \(\vec v\) parallel to the direction of \(\vec u\)

3 Dot Product: Scalar Projections

component functions

3 Line Integrals: Line Integrals of Vectors

component form

1 Vectors: Parametric Lines

component or scalar projection of \(\vec v\) parallel to \(\vec u\)

1 Vectors: Scalar Projections

component or scalar projection of the vector \(\vec v\) parallel to the direction of \(\vec u\)

1 Vectors: Scalar Projections

component or scalar projection of the vector \(\vec v\) perpendicular to the direction of \(\vec u\)

3 Dot Product: Scalar Projections

component or scalar projection of \(\vec v\) perpenducular to \(\vec u\)

3 Dot Product: Scalar Projections

component form

3 Lines and Planes: Lines in \(\mathbb{R}^2\) - More on the Normal Equation

component or scalar projection of the vector \(\vec v\) perpendicular to the direction of \(\vec u\)

1 Vectors: Scalar Projections

component or scalar projection of \(\vec v\) parallel to \(\vec u\)

3 Dot Product: Scalar Projections

components

1 Vectors: Scalar Projections

components

3 Dot Product: Scalar Projections

composition of functions

3 Chain Rule: Composition of Functions

composition of functions

1 Chain Rule & Implicit Differentiation: Compositions

compute an integral which is improper at one or more interior points

2 Improper Integrals: Integrals Improper at an Interior Point

compute an integral which is improper at a finite left endpoint

2 Improper Integrals: Integrals Improper at a Finite Endpoint

compute an integral which is improper at positive infinity

2 Improper Integrals: Integrals Improper at an Infinite Endpoint

conditionally convergent

2 Convergence of General Series: The Absolute Convergence Test

cone

3 Curves and Surfaces: Quadratic Surfaces - Part 1

conic sections

3 Curves and Surfaces: undefined

conjugate

2 Sequences: Indeterminate Form \(\infty-\infty\)

conjugate

1 Sequences: Indeterminate Form \(\infty-\infty\)

conservative

3 Divergence, Curl and Potentials: Non-Existence of a Scalar Potential

Constant Sequences

1 Computing Limits: Limits of Special Functions

1 Precise Limits and Continuity: Limit Laws

constant of integration

2 Differential Equations and Initial Value Problems: Separable Differential Equations --- Example 7

2 Riemann Sums: Indefinite Integrals

Constant Sequences

2 Sequences: Special Limits

constant

2 Series: Geometric Series - Finite

constants of integration

2 Differential Equations and Initial Value Problems: Solutions of Differential Equations

constraint equation

3 Max-Min Problems: Constrained Max-Min Problems

continuous from the right

3 Limits and Continuity: Intuitive Limits for Functions of 1 Variable (Review)

continuous on a region

3 Limits and Continuity: Continuity

continuous from the left

3 Limits and Continuity: Intuitive Limits for Functions of 1 Variable (Review)

continuous from the left

1 Intuitive Limits and Continuity: 2-Sided Limits and Continuity

continuous

3 Limits and Continuity: Continuity

continuous from the right

1 Intuitive Limits and Continuity: 2-Sided Limits and Continuity

continuous

1 Intuitive Limits and Continuity: 2-Sided Limits and Continuity

contour

3 Functions of Several Variables: Level Sets and Contour Plots

contour plot

3 Functions of Several Variables: Level Sets and Contour Plots

contractable

3 Divergence, Curl and Potentials: Non-Existence of a Vector Potential

contracting

3 Divergence, Curl and Potentials: Algebraic Definition of Divergence

contraction of the fluid

3 Line Integrals: Expansion and Contraction

Convergence Test

2 Convergence of General Series: Exercises

2 Convergence of Positive Series:

convergent

1 Precise Limits and Continuity: Precise Limits at Infinity

convergent

2 Sequences: Limits - Definition

converges

1 Intuitive Limits and Continuity: Left and Right Limits

converges to

2 Sequences: Limits - Definition

converges

3 Limits and Continuity: Intuitive Limits for Functions of 1 Variable (Review)

converges to

1 Precise Limits and Continuity: Precise Limits at Infinity

converges to \(S\)

2 Series: The Sum of a Series

converging

3 Divergence, Curl and Potentials: Algebraic Definition of Divergence

Converting Between Rectangular and Polar Coordinates

2 Polar Coordinates: Converting Between Rectangular and Polar Coordinates

Converting Between Rectangular and Polar Coordinates

3 Coordinate Systems: 2D Converting between Rectangular and Polar

convex hull

3 Multiple Integrals in Curvilinear Coordinates: Polar Coordinates - Applications

coordinate curve

3 Coordinate Systems: 2D Polar Coordinates

coordinate dimensions

3 Multiple Integrals in Curvilinear Coordinates: Polar Coordinates - Grid Cell

coordinate curves

3 Parametric Surfaces and Surface Integrals: Coordinate Curves and Tangent Vectors

coordinate tangent vectors,

3 Multiple Integrals in Curvilinear Coordinates: 2D Curvilinear Coordinates - Grid Cell

coordinate surface

3 Coordinate Systems: 3D Cylindrical Coordinates

coordinate grid

3 Coordinate Systems: 2D Polar Coordinates

coordinate center

3 Multiple Integrals in Curvilinear Coordinates: Polar Coordinates - Grid Cell

coordinates

3 Curves and Surfaces: Surfaces in \(\mathbb{R}^3\)

3 Parametric Surfaces and Surface Integrals: Position

Co-Planar Vectors

3 Cross Product: Geometric Interpretation of Triple Products

Coulomb's Law of Electrostatics

3 Divergence, Curl and Potentials: The Divergence of a Gradient (\(\vec\nabla\cdot\vec\nabla f\)) - The Laplacian

counterclockwise

3 Stokes' Theorem: Surfaces whose Boundaries have Several Curves

critical points

3 Functions of Several Variables: Functions of \(2\) Variables

3 Max-Min Problems: Local Minima, Local Maxima and Inflection Points of Functions of \(1\) Variable

critical point

3 Functions of Several Variables: Functions of \(1\) Variable

critical points

1 Max/Min Problems: \(2^\text{nd}\) Derivative Test

cross

3 Curves and Surfaces: Quadratic Curves - Part 1

cross product

3 Cross Product: Algebraic Definition of Cross Products

cubic

2 Differential Equations and Initial Value Problems: What is a Differential Equation?

curl

3 Divergence, Curl and Potentials: undefined

Curl of a Gradient Identity

3 Divergence, Curl and Potentials: The Curl of a Gradient (\(\vec\nabla\times\vec\nabla f\))

curl field

3 Divergence, Curl and Potentials: Non-Existence of a Vector Potential

curl-free

3 Divergence, Curl and Potentials: Algebraic Definition of Curl

curly d

3 Partial Derivatives and Tangent Planes: Computation

curvature

3 Properties of Curves: Curvature

curve

1 Vectors: Parametric Curves

curve

3 Curves and Surfaces: Curves in \(\mathbb{R}^2\)

curvilinear rectangle

3 Multiple Integrals in Curvilinear Coordinates: 2D Curvilinear Coordinates - Grid Cell Area

curvilinear coordinate system

3 Parametric Surfaces and Surface Integrals: Introduction to Parametric Surfaces

curvilinear coordinates

3 Multiple Integrals in Curvilinear Coordinates: 2D Curvilinear Coordinates - Grid Cell

cusp

1 Vectors: Parametric Curves

cusp

2 Arc Length and Surface Area: Arc Length of a Parametric Curve

cusp

3 Curves and Surfaces: Curves in \(\mathbb{R}^2\)

cylinder

3 Curves and Surfaces: Quadratic Surfaces - Part 2

Cylindrical Coordinates

3 Coordinate Systems: 3D Cylindrical Coordinates

cylindrical shells

2 Volume: Solids of Revolution

decreasing

2 Sequences: Monotonic Sequences

decreasing in absolute value

2 Convergence of General Series: The Alternating Series Test

defined recursively

2 Sequences: Recursively Defined Sequences

(definite) integral

2 Riemann Sums: Riemann Sums and Areas

degree

2 Differential Equations and Initial Value Problems: What is a Differential Equation?

-degree term

2 Power Series: What is a Power Series?

del operator

3 Divergence, Curl and Potentials: undefined

"del"

3 Partial Derivatives and Tangent Planes: Algebraic Definition of Partial Derivatives

"del".

3 Directional Derivatives and Gradients: Derivative Along a Curve and the Gradient

dependent variable

2 Differential Equations and Initial Value Problems: What is a Differential Equation?

dependent

3 Chain Rule: Chain Rule with \(1\) Intermediate Variable

dependent

1 Chain Rule & Implicit Differentiation: Chain Rule

dependent variable

3 Functions of Several Variables: Formulas

derivative of \(f(x,y)\) along \(\vec v\)

3 Directional Derivatives and Gradients: Derivative Along a Vector

3 Partial Derivatives and Tangent Planes: Algebraic Definition of Partial Derivatives

Derivative of the Area Function

2 Antiderivatives: Area as Antiderivative

Derivative along a Curve

3 Directional Derivatives and Gradients: Derivative Along a Vector

derivative of the function \(f\) along the curve \(\vec r(t)\)

3 Directional Derivatives and Gradients: Derivative Along a Curve and the Gradient

Derivative as a Function

1 Definitions and Tangent Lines: Derivative as a Function

derivative of the function \(f(x)\)

1 Definitions and Tangent Lines: Derivative as a Function

Derivative of the Integral

2 Riemann Sums: The Fundamental Theorem of Calculus for Integrals

Derivative of the Integral

1 Riemann Sums: The Fundamental Theorem of Calculus for Integrals

derivative of a vector function

3 Properties of Curves: Derivatives of Vector Functions

Derivative along a Vector

3 Directional Derivatives and Gradients: Derivative Along a Vector

Derivative at a Point

1 Definitions and Tangent Lines: Derivative at a Point

derivative of the function \(f(x)\) at the point \(x=a\)

1 Definitions and Tangent Lines: Derivative at a Point

derivative of a vector function

1 Parametric Curves: Velocity Vector

Derivative of the Area Function

1 Antiderivatives: Area as an Antiderivative

determinant of a \(3\times3\) matrix

3 Cross Product: Method of Diagonals

determinant of a \(2\times2\) matrix

3 Cross Product: Method of Diagonals

difference (or change) in the independent variable

3 Linear Approximation: Differentials with \(1\) Variable

difference (or change) in the dependent variable

1 Differentials & Linear Approximation: Differences

difference (or change) in the dependent variable

3 Linear Approximation: Differentials with \(1\) Variable

difference (or change) in the independent variable

1 Differentials & Linear Approximation: Differences

difference (or change) in the independent variables

3 Linear Approximation: Differentials with \(2\) Variables

difference (or change) in the dependent variable

3 Linear Approximation: Differentials with \(2\) Variables

differential equation

1 Antiderivatives: From Acceleration to Velocity and Position

differential of the dependent variable

1 Differentials & Linear Approximation: Differences

differential of arc length

2 Arc Length and Surface Area: Differential of Arc Length

differential notation

3 Higher Order Derivatives: Higher Derivatives

differential of the independent variable

3 Linear Approximation: Differentials with \(1\) Variable

differential of the dependent variable

3 Linear Approximation: Differentials with \(1\) Variable

differential of the independent variable

1 Differentials & Linear Approximation: Differences

differential equation

2 Antiderivatives: Differential Equations and Initial Value Problems

differentials of the independent variables

3 Linear Approximation: Differentials with \(2\) Variables

Differentiation of Power Series

2 Power Series: Differentiation

direct second partials

3 Higher Order Derivatives: Higher Derivatives

direction

3 Vectors: Vectors as Arrows

direction

1 Vectors: Vectors as Arrows

direction cosines

3 Vectors: Direction

direction cosines

1 Vectors: Direction

direction angles

3 Vectors: Direction

direction angles

1 Vectors: Direction

directional derivative of \(f(x,y)\) along \(\vec v\)

3 Directional Derivatives and Gradients: Directional Derivatives

directional derivative of \(f(x,y)\) in the direction of \(\vec v\)

3 Directional Derivatives and Gradients: Directional Derivatives

discriminant

3 Max-Min Problems: Local Minima, Local Maxima and Saddle Points of Functions of \(2\) Variables

displacement vector

1 Vectors: Displacements

displacement

1 Vectors: Displacements

displacement vector

3 Vectors: The Vector Between Two Points - Displacement

displacement

3 Vectors: The Vector Between Two Points - Displacement

Distance Formula in \(\mathbb{R}^3\)

3 Coordinate Systems: Distance Formula in 3D

Distance Formula in \(\mathbb{R}^n\)

3 Coordinate Systems: Distance Formula in Higher Dimensions (Optional)

Distance from a Point to a Line in \(\mathbb{R}^2\)

3 Lines and Planes: Distance from a Point to a Line - \(\mathbb{R}^2\)

Distance from a Point to a Line

3 Lines and Planes: Distance from a Point to a Line - \(\mathbb{R}^3\)

Distance Formula

3 Coordinate Systems: Distance Formula in 2D

Distance from a Point to a Plane

3 Lines and Planes: Distance from a Point to a Plane

Distance from a Point to a Line in \(\mathbb{R}^3\)

3 Lines and Planes: Distance from a Point to a Line - \(\mathbb{R}^3\)

divergence

3 Divergence, Curl and Potentials: undefined

Divergence of a Curl Identity

3 Divergence, Curl and Potentials: The Divergence of a Curl (\(\vec\nabla\cdot\vec\nabla\times\vec F\))

divergence-free

3 Divergence, Curl and Potentials: Algebraic Definition of Divergence

divergent

1 Precise Limits and Continuity: Precise Limits at Infinity

divergent

2 Convergence of General Series: The Absolute Convergence Test

2 Sequences: Overview

2 Series: Limits - Definition

diverges

1 Intuitive Limits and Continuity: Left and Right Limits

1 Precise Limits and Continuity: Precise Limits at Infinity

diverges to \(-\infty\)

3 Limits and Continuity: When Limits Do Not Exist

diverges to \(\infty\)

3 Limits and Continuity: When Limits Do Not Exist

diverges to \(-\infty\) on the left or right

3 Limits and Continuity: When Limits Do Not Exist

diverges

2 Sequences: Limits - Definition

2 Series: Precise Definiton of Limits

diverges to \(\infty\) on the left or right

3 Limits and Continuity: When Limits Do Not Exist

diverges to \(-\infty\)

1 Intuitive Limits and Continuity: 2-Sided Limits and Continuity

diverges

3 Limits and Continuity: Intuitive Limits for Functions of 1 Variable (Review)

diverges to \(\infty\) on the left or right

1 Intuitive Limits and Continuity: When Limits Do Not Exist

diverges to positive infinity

2 Sequences: Limits - Definition

diverges to \(\infty\)

1 Intuitive Limits and Continuity: 2-Sided Limits and Continuity

diverges to \(-\infty\) on the left or right

1 Intuitive Limits and Continuity: When Limits Do Not Exist

diverges to negative infinity

2 Sequences: Limits - Definition

diverging

3 Divergence, Curl and Potentials: Algebraic Definition of Divergence

dividend

2 Partial Fractions: General Decompositions

divisor

2 Partial Fractions: General Decompositions

does not have a sum

2 Series: The Sum of a Series

dog saddle

3 Functions of Several Variables: Functions of \(2\) Variables

domain

3 Functions of Several Variables: Domain and Image

dot product

1 Vectors: Dot Product - Algebraic

dot product

3 Dot Product: Algebraic Definition

double integral of \(f(x,y)\) over the region \(R\)

3 Multiple Integrals as Riemann Sums: Riemann Sum Definition of Double Integrals

drag coefficient

2 Applications of Differential Equations: Falling with Air Resistance

electric charge

2 Mass and Center of Mass: Electric Charge - Bar with Non-Uniform Charge Density

electrostatic potential energy

3 Fundamental Theorem of Calculus for Curves: Coulomb Electrostatic Force

Eliminate a Variable Method

3 Max-Min Problems: Eliminate a Variable Method

ellipse

3 Curves and Surfaces: Quadratic Curves - Part 1

ellipsoid

3 Curves and Surfaces: Quadratic Surfaces - Part 1

elliptic cylinders

3 Curves and Surfaces: Quadratic Surfaces - Part 2

elliptic polar coordinate system

3 Multiple Integrals in Curvilinear Coordinates: 2D Curvilinear Coordinates - Ellipse Example

elliptic paraboloid

3 Curves and Surfaces: Quadratic Surfaces - Part 2

equal to

1 Precise Limits and Continuity: Precise Limits at Infinity

equal to

3 Limits and Continuity: Precise Limits for Functions of \(1\) Variable (Optional Review)

equal to

2 Sequences: Precise Definiton of Limits

Equation of a Tangent Plane to a Graph

3 Partial Derivatives and Tangent Planes: Comparison of Tangent Line and Tangent Plane Derivations

Equation of a Circle

3 Coordinate Systems: Distance Formula in 2D

Equation of a Sphere

3 Coordinate Systems: Distance Formula in 3D

Equation of a Tangent Line to a Graph

1 Definitions and Tangent Lines: Tangent Lines

Equation of a Tangent Line to a Graph

3 Partial Derivatives and Tangent Planes: Tangent Line to the Graph of a Function

Equation of a Tangent HyperPlane to a Graph

3 Partial Derivatives and Tangent Planes: More Variables

Equation of a Sphere in \(\mathbb{R}^n\)

3 Coordinate Systems: Distance Formula in Higher Dimensions (Optional)

Equivalent Form

1 Precise Limits and Continuity: Limit Laws

error in the approximation

2 Convergence of Positive Series: Approximating the Series

evaluation point

3 Multiple Integrals as Riemann Sums: Riemann Sum Definition of Single Integrals (Review)

evaluation point

2 Riemann Sums: Riemann Sums and Integrals (Optional)

evaluation point

1 Riemann Sums: Riemann Sums and Integrals

exists

2 Sequences: Precise Definiton of Limits

exists

3 Limits and Continuity: Precise Limits for Functions of \(1\) Variable (Optional Review)

exists

1 Precise Limits and Continuity: Precise Limits at Infinity

expanded

2 Power Series: What is a Power Series?

expanding

3 Divergence, Curl and Potentials: Algebraic Definition of Divergence

expansion of the fluid

3 Line Integrals: Expansion and Contraction

expansion of the vector field

3 Line Integrals: Expansion and Contraction

expansion by minors

3 Cross Product: Expansion by Minors on a Row or Column

expected value

3 Line Integrals: Average Value

explicit function definition

3 Chain Rule: Implicit Functions

explicit form of the solution

2 Differential Equations and Initial Value Problems: Separable Differential Equations --- Example 2

explicit function definition

1 Chain Rule & Implicit Differentiation: Implicit Functions

Exponential Sequences

1 Computing Limits: Limits of Special Functions

1 Precise Limits and Continuity: Limits of Non-Linear Functions

Exponential Sequences

2 Sequences: Special Limits

Extended Mean Value Theorem

2 Taylor Series: Taylor Remainder

extrema

3 Functions of Several Variables: Functions of \(1\) Variable

3 Max-Min Problems: Functions of \(2\) Variables

extremal

3 Max-Min Problems: Constrained Max-Min Problems

extreme cases

2 Convergence of Positive Series: The Limit Comparison Test - Proof

extremize

3 Max-Min Problems: Constrained Max-Min Problems

finite

2 Sequences: Definition

Finite Geometric Series

2 Series: Geometric Series - Finite

finite geometric series

2 Series: Geometric Series - Finite

finite

1 Computing Limits: Limits at Infinity

finite series

2 Series: Definition

first moment of the mass

2 Mass and Center of Mass: Center of Mass - Bar with Non-Uniform Density

First Derivative Test

3 Max-Min Problems: Local Minima, Local Maxima and Inflection Points of Functions of \(1\) Variable

first moments of the mass

3 Line Integrals: Mass, Center of Mass and Centroid

first moments of arc length

3 Line Integrals: Mass, Center of Mass and Centroid

first term

2 Series: Geometric Series - Finite

First Derivative Test

1 Max/Min Problems: \(2^\text{nd}\) Derivative Test

flow diagram

2 Differential Equations and Initial Value Problems: Separable Differential Equations --- Example 2

flow of the fluid

3 Line Integrals: Flow and Circulation

flow of the vector field

3 Line Integrals: Flow and Circulation

for Manipulating the Hand.

3 Cross Product: Right Hand Rule

force

3 Vectors: Application: Balancing Forces

force

1 Vectors: Balancing Forces

free scalar

3 Vectors: Free vs. Located

free vector

3 Vectors: Free vs. Located

free fall

2 Applications of Differential Equations: Free Fall

Frenet Equations

3 Properties of Curves: Frenet Frames & Derivatives of TNB

front

3 Parametric Surfaces and Surface Integrals: Vector Surface Integrals

frustrum of a cone

2 Arc Length and Surface Area: Area of a Surface of Revolution

frustum of a cone

2 Arc Length and Surface Area: Area of a Surface of Revolution

Fubini's Theorem: Integral over a Polar Rectangle

3 Multiple Integrals in Curvilinear Coordinates: Polar Coordinates - Integral over a Polar Rectangle

Fubini's Theorem

3 Multiple Integrals in Curvilinear Coordinates: Polar Coordinates - Integral over a Polar Rectangle - Derivation

3 Iterated Integrals: Computing Iterated Integrals with Constant Limits

3 Multiple Integrals as Riemann Sums: Double Integrals over a Rectangle

function

3 Functions of Several Variables: Formulas

Fundamental Theorem of Calculus for Curves (FTCC)

3 Fundamental Theorem of Calculus for Curves: undefined

Gauss' Theorem without Holes

3 Gauss' Theorem: The Theorem

Gauss' Theorem

3 Gauss' Theorem: undefined

general antiderivative

1 Antiderivatives: Non-Uniqueness of Antiderivatives

general

2 Convergence of Positive Series: Overview

general antiderivative

2 Antiderivatives: Non-Uniqueness of Antiderivatives

general series

2 Convergence of General Series: Overview

Geometric Formula for Dot Product

1 Vectors: Dot Product - Geometric

Geometric Formula for Dot Product

3 Dot Product: Geometric Formula

Geometric Description of the Cross Product

3 Cross Product: Geometric Interpretation of Cross Products

geometric series

2 Series: Geometric Series - Finite

Geometric Interpretation of the Triple Product

3 Cross Product: Geometric Interpretation of Triple Products

geometrical domains

3 Parametric Surfaces and Surface Integrals: Introduction to Parametric Surfaces

global maximum

3 Functions of Several Variables: Functions of \(1\) Variable

3 Max-Min Problems: Functions of \(2\) Variables

global minimum

3 Functions of Several Variables: Functions of \(1\) Variable

3 Max-Min Problems: Functions of \(2\) Variables

gradient

3 Divergence, Curl and Potentials: undefined

3 Higher Order Derivatives: The Del Operator

3 Directional Derivatives and Gradients: The Hessian

gradient field

3 Divergence, Curl and Potentials: Non-Existence of a Scalar Potential

graph of the parametric curve

3 Curves and Surfaces: Curves in \(\mathbb{R}^2\)

graph of an equation in \(3\) variables

3 Functions of Several Variables: Graphs of Equations

graph of a function of \(2\) variables

3 Functions of Several Variables: Graphs of Functions

graph of the parametric surface

3 Curves and Surfaces: Surfaces in \(\mathbb{R}^3\)

graph of an equation in \(2\) variables

3 Functions of Several Variables: Graphs of Equations

graph of the parametric curve

1 Vectors: Parametric Curves

graph of a function of \(3\) variables

3 Functions of Several Variables: Graphs of Functions

graph of a function of \(1\) variable

3 Functions of Several Variables: Graphs of Functions

gravitational potential energy

3 Fundamental Theorem of Calculus for Curves: Gravitational Force near the Earth

Green's Theorem

3 Green's Theorem: undefined

Green's Theorem without Holes

3 Green's Theorem: The Theorem

grid cell

3 Multiple Integrals in Curvilinear Coordinates: 2D Curvilinear Coordinates - Grid Cell Area

growth constant

2 Differential Equations and Initial Value Problems: Separable Differential Equations --- Example 7

harmonic series

2 Convergence of Positive Series: The Integral Test

harmonic function

3 Divergence, Curl and Potentials: The Divergence of a Gradient (\(\vec\nabla\cdot\vec\nabla f\)) - The Laplacian

has sum \(S\)

2 Series: The Sum of a Series

has a limit

2 Sequences: Limits - Definition

\(\hat\imath\,\hat\jmath\,\hat k\) notation

3 Vectors: \(ijk\) Notation

\(\hat\imath\,\hat\jmath\,\hat k\) notation

1 Vectors: \(\hat\imath\hat\jmath\) Notation

Heat Equation

3 Divergence, Curl and Potentials: The Divergence of a Gradient (\(\vec\nabla\cdot\vec\nabla f\)) - The Laplacian

helicity

3 Properties of Curves: Torsion

helix

2 Arc Length and Surface Area: Parametric Curves

Hessian

3 Higher Order Derivatives: The Hessian

horizontal axis

3 Coordinate Systems: 2D Rectangular Coordinates

Horizontal and Vertical Tangents to a Parametric Curve

2 Arc Length and Surface Area: Horizontal and Vertical Tangents

horizontal inflection points

3 Functions of Several Variables: Functions of \(1\) Variable

horizontal inflection point

1 Max/Min Problems: \(2^\text{nd}\) Derivative Test

horizontal inflection point

3 Max-Min Problems: Local Minima, Local Maxima and Inflection Points of Functions of \(1\) Variable

hyperbola

3 Curves and Surfaces: Quadratic Curves - Part 1

hyperbolic paraboloid

3 Curves and Surfaces: Quadratic Surfaces - Part 2

hyperbolic cylinders

3 Curves and Surfaces: Quadratic Surfaces - Part 2

hyperboloid of one sheet

3 Curves and Surfaces: Quadratic Surfaces - Part 1

hyperboloid of two sheets

3 Curves and Surfaces: Quadratic Surfaces - Part 1

Identity Sequence

1 Computing Limits: Limits of Special Functions

1 Precise Limits and Continuity: Limit Laws

Identity Sequence

2 Sequences: Special Limits

\(ij\)-entry in the Hessian

3 Higher Order Derivatives: The Hessian

image

3 Functions of Several Variables: Domain and Image

implicit function definition

1 Chain Rule & Implicit Differentiation: Implicit Functions

implicit function definition

3 Chain Rule: Implicit Functions

implicit form of the solution

2 Differential Equations and Initial Value Problems: Separable Differential Equations --- Example 2

Improper Integrals of Inverse Powers

2 Improper Integrals: Relation between \(\displaystyle\int\dfrac{1}{x^p}\,dx\) and \(\displaystyle\int\dfrac{1}{\sqrt[p]{x}}\,dx\)

Improved Comparison Bound on the Error in an Approximation

2 Convergence of Positive Series: Improving the Approximation Using Comparisons

Improved Integral Bound on the Error in an Approximation

2 Convergence of Positive Series: Improving the Approximation Using Integrals

Improved Alternating Series Bound on the Error in an Approximation

2 Convergence of General Series: Improving the Approximation for an Alternating Series

In summary:

3 Multiple Integrals in Curvilinear Coordinates: 2D Curvilinear Coordinates - The Jacobian

incompressible

3 Divergence, Curl and Potentials: Algebraic Definition of Divergence

increasing

2 Sequences: Monotonic Sequences

indefinite integral

2 Riemann Sums: Indefinite Integrals

independent

3 Chain Rule: Chain Rule with \(1\) Intermediate Variable

independent

1 Chain Rule & Implicit Differentiation: Chain Rule

independent variable

2 Differential Equations and Initial Value Problems: What is a Differential Equation?

independent variables

3 Functions of Several Variables: Formulas

indeterminate forms

1 Computing Limits: Limit Laws

indeterminate forms

2 Sequences: Limit Laws

indirect

3 Higher Order Derivatives: Higher Derivatives

infinite

2 Sequences: Definition

infinite series

2 Series: Definition

infinite

1 Computing Limits: Limits at Infinity

Infinite Limit of a Sequence

2 Sequences: Limits - Definition

infinite geometric series

2 Series: Geometric Series - Infinite

Infinite Geometric Series

2 Series: Geometric Series - Infinite

Infinite Limit from the Right

1 Precise Limits and Continuity: Precise Infinite Limits

initial value problem

2 Antiderivatives: Differential Equations and Initial Value Problems

initial terms

2 Sequences: Recursively Defined Sequences

initial value problem

1 Antiderivatives: From Acceleration to Velocity and Position

inner product

1 Vectors: Dot Product - Algebraic

inner

3 Chain Rule: Chain Rule with \(1\) Intermediate Variable

inner

1 Chain Rule & Implicit Differentiation: Chain Rule

inner product

3 Dot Product: Algebraic Definition

input tolerance

3 Limits and Continuity: Precise Limits for Functions of \(1\) Variable (Optional Review)

input tolerance

1 Precise Limits and Continuity: Precise Infinite Limits

integral converges

2 Improper Integrals: Integrals Improper at a Finite Endpoint

integral of \(f(x)\) from \(x=a\) to \(x=b\)

3 Multiple Integrals as Riemann Sums: Riemann Sum Definition of Single Integrals (Review)

integral

1 Riemann Sums: Riemann Sums and Areas

Integral over a Type \(x\) Region

3 Multiple Integrals in Rectangular Coordinates: Double Integrals over a Non-Rectangular Region - Derivations

integral function for \(f(x)\)

1 Riemann Sums: The Fundamental Theorem of Calculus for Integrals

Integral over a Type \(r\) Region

3 Multiple Integrals in Curvilinear Coordinates: Polar Coordinates - Integral over a Non-Rectangular Region

integral function for \(f(x)\)

2 Riemann Sums: The Fundamental Theorem of Calculus for Integrals

Integral over a Type \(\theta\) Region

3 Multiple Integrals in Curvilinear Coordinates: Polar Coordinates - Integral over a Non-Rectangular Region

Integral over a Type \(y\) Region

3 Multiple Integrals in Rectangular Coordinates: Double Integrals over a Non-Rectangular Region - Derivations

integral bound on the error in the approximation

2 Convergence of Positive Series: Approximating the Series - Integral Bounds

integral diverges

2 Improper Integrals: Integrals Improper at a Finite Endpoint

integrate a differential equation

2 Differential Equations and Initial Value Problems: Solutions of Differential Equations

Integration by Parts

2 Integration by Parts: Definite Integrals

Integration by Substitution

2 Integration by Substitution: Indefinite Integrals

Integration by Substitution

1 Integration by Substitution: Indefinite Integrals

Integrations of Power Series

2 Power Series: Integration

intermediate

1 Chain Rule & Implicit Differentiation: Chain Rule

intermediate

3 Chain Rule: Chain Rule with \(1\) Intermediate Variable

Interpretation of an Integral as Net Area

2 Riemann Sums: Riemann Sums and Integrals (Optional)

Interpretation of Integral as Net Area

1 Riemann Sums: Riemann Sums and Integrals

interval of convergence

2 Power Series: Sequences and Series of Functions

interval of convergence

2 Power Series: Interval of Convergence

inverse

2 Partial Fractions: Complex Methods (Optional)

irreducible

2 Partial Fractions: General Decompositions

irrotational

3 Divergence, Curl and Potentials: Algebraic Definition of Curl

is convergent

2 Series: The Sum of a Series

iterated double integral

3 Iterated Integrals: Computing Iterated Integrals with Constant Limits

iterated integral

3 Iterated Integrals: Computing Iterated Integrals with Constant Limits

iterated triple integral

3 Iterated Integrals: Computing Iterated Integrals with Constant Limits

Jacobian determinant

3 Multiple Integrals in Curvilinear Coordinates: 3D Curvilinear Coordinates - The Jacobian

Jacobian factor

3 Multiple Integrals in Curvilinear Coordinates: 2D Curvilinear Coordinates - Grid Cell Area

Jacobian Factor:

3 Multiple Integrals in Curvilinear Coordinates: 2D Curvilinear Coordinates - The Jacobian

Jacobian Matrix:

3 Multiple Integrals in Curvilinear Coordinates: 2D Curvilinear Coordinates - The Jacobian

Jacobian Determinant:

3 Multiple Integrals in Curvilinear Coordinates: 2D Curvilinear Coordinates - The Jacobian

\(k^\text{th}\) moment of the mass

2 Mass and Center of Mass: Center of Mass - Bar with Non-Uniform Density

\(k^\text{th}\) partial sum

2 Series: The Sum of a Series

\(k^\text{th}\) degree Taylor polynomial

3 Higher Order Derivatives: Taylor Series and Polynomials for \(2\) Variable

Lagrange equations

3 Max-Min Problems: Lagrange Multiplier Method

Lagrange Multiplier Method

3 Max-Min Problems: Multiple Constraint Problems (Optional)

Lagrange multiplier

3 Max-Min Problems: Lagrange Multiplier Method

Laplace Equation

3 Divergence, Curl and Potentials: The Divergence of a Gradient (\(\vec\nabla\cdot\vec\nabla f\)) - The Laplacian

Laplacian

3 Divergence, Curl and Potentials: The Divergence of a Gradient (\(\vec\nabla\cdot\vec\nabla f\)) - The Laplacian

Law of Cosines

3 Dot Product: Geometric Formula

Law of Cosines

1 Vectors: Dot Product - Geometric

leading principle minor determinants

3 Max-Min Problems: Second Derivative Test with More Variables (Optional)

Left Riemann Sum Error and Bound Theorem

2 Numerical Integration: Error Bounds - Left and Right Riemann Sums

left handed system

3 Cross Product: Geometric Interpretation of Triple Products

left or right limit is positive infinity

1 Intuitive Limits and Continuity: When Limits Do Not Exist

left or right limit is negative infinity

3 Limits and Continuity: When Limits Do Not Exist

left or right limit is negative infinity

1 Intuitive Limits and Continuity: When Limits Do Not Exist

left or right limit is positive infinity

3 Limits and Continuity: When Limits Do Not Exist

left or right limit is oscillitory divergent

1 Intuitive Limits and Continuity: When Limits Do Not Exist

left or right limit is oscillitory divergent

3 Limits and Continuity: When Limits Do Not Exist

length

1 Vectors: Magnitude

length

2 Partial Fractions: Complex Methods (Optional)

length

3 Vectors: Magnitude

level set

3 Functions of Several Variables: Level Sets and Contour Plots

level surfaces

3 Functions of Several Variables: Functions of \(3\) Variables

level curves

3 Functions of Several Variables: Functions of \(2\) Variables

level point

3 Functions of Several Variables: Functions of \(1\) Variable

limit of the function, \(f(x)\), as \(x\) approaches \(a\) from the left is infinity

1 Precise Limits and Continuity: Precise Infinite Limits

limit of the function, \(f(x)\), as \(x\) approaches \(a\) from the right

3 Limits and Continuity: Intuitive Limits for Functions of 1 Variable (Review)

limit of the function, \(f(x)\), as \(x\) approaches \(a\) from the right

1 Intuitive Limits and Continuity: Left and Right Limits

limit of the function, \(f(x)\), as \(x\) approaches \(a\) from the left

1 Intuitive Limits and Continuity: Left and Right Limits

limit of the function, \(f(x)\), as \(x\) approaches \(a\) from the left

3 Limits and Continuity: Intuitive Limits for Functions of 1 Variable (Review)

limit of the \(n^\text{th}\) root of the \(n^\text{th}\) term

2 Convergence of Positive Series: Overview

limit of the ratio of successive terms

2 Convergence of Positive Series: Overview

limit of a vector function

3 Properties of Curves: Limits of Vector Functions

Limit Laws

3 Limits and Continuity: Limit Laws

limit

2 Sequences: Precise Definiton of Limits

Limit Laws

1 Computing Limits: Limit Laws

1 Precise Limits and Continuity:

Limit of a Function from the Right

1 Precise Limits and Continuity: Precise Left and Right Limits

limit

3 Limits and Continuity: Intuitive Limits for Functions of 1 Variable (Review)

limit of the function, \(f(x)\), as \(x\) approaches \(a\) from the right

1 Precise Limits and Continuity: Precise Left and Right Limits

limit of the absolute value of the ratio of successive terms

2 Convergence of General Series: Overview

limit

1 Intuitive Limits and Continuity: 2-Sided Limits and Continuity

1 Precise Limits and Continuity: Precise Limits at Infinity

limit from the right

2 Improper Integrals: Integrals Improper at a Finite Endpoint

Limit of a Sequence

2 Sequences: Limits - Definition

limit of the function, \(f(x)\), as \(x\) approaches \(a\) from the right is infinity

1 Precise Limits and Continuity: Precise Infinite Limits

limit of the function, \(f(x)\), as \(x\) approaches \(a\) from the right is negative infinity

1 Precise Limits and Continuity: Precise Infinite Limits

limit of the \(n^\text{th}\) root of the absolute value of the \(n^\text{th}\) term

2 Convergence of General Series: Overview

limit of the function, \(f(x)\), as \(x\) approaches \(a\) from the left is negative infinity

1 Precise Limits and Continuity: Precise Infinite Limits

Limit Laws

2 Sequences: Limit Laws

Limit of a Function from the Right

3 Limits and Continuity: Precise Limits for Functions of \(1\) Variable (Optional Review)

limit of the function, \(f(x)\), as \(x\) approaches \(a\) is infinity

1 Precise Limits and Continuity: Precise Infinite Limits

limit of the function, \(f(x)\), as \(x\) approaches \(a\) from the right

3 Limits and Continuity: Precise Limits for Functions of \(1\) Variable (Optional Review)

limit of the function, \(f(x,y)\), as \((x,y)\) approaches \((a,b)\)

3 Limits and Continuity: Precise Limits for Functions of \(2\) Variables (Optional)

limit of the function, \(f(x)\), as \(x\) approaches \(a\) is negative infinity

1 Precise Limits and Continuity: Precise Infinite Limits

Limit of a Function of \(2\) Variables

3 Limits and Continuity: Precise Limits for Functions of \(2\) Variables (Optional)

limit of a difference quotient

1 Definitions and Tangent Lines: Derivative at a Point

Limit of a Sequence

1 Precise Limits and Continuity: Precise Limits at Infinity

limit is \(-\infty\)

2 Sequences: Limits - Definition

limit is \(+\infty\)

2 Sequences: Limits - Definition

Limits of Functions of \(2\) Variables

3 Limits and Continuity: Precise Limits for Functions of \(2\) Variables (Optional)

Limits of Functions of Two Variables

3 Limits and Continuity: Intuitive Limits for Functions of 2 Variables

line integral of a vector field

3 Line Integrals: Line Integrals of Vectors

line

3 Lines and Planes: Lines in \(\mathbb{R}^2\) - More on the Normal Equation

line integral of a function

3 Line Integrals: Line Integrals of Scalars

line

1 Vectors: Parametric Lines

line of longitude

3 Coordinate Systems: 3D Spherical Coordinates

line of latitude

3 Coordinate Systems: 3D Spherical Coordinates

linear combination

3 Vectors: Linear Combinations

linear combination

1 Vectors: Linear Combinations

linear

2 Differential Equations and Initial Value Problems: What is a Differential Equation?

Linear Approximation with \(1\) Variable

3 Linear Approximation: Linear Approximation with \(1\) Variable

Linear Approximation

1 Differentials & Linear Approximation: Linear Approximation

Linear Approximation with \(2\) Variables

3 Linear Approximation: Linear Approximation with \(2\) Variables

Linear Approximation with \(3\) Variables

3 Linear Approximation: Linear Approximation with \(3\) or more Variables

linear mass density

3 Multiple Integrals in Rectangular Coordinates: Review of Center of Mass of a Bar

linear density

2 Mass and Center of Mass: Total Mass

linear electric charge density

2 Mass and Center of Mass: Electric Charge - Bar with Non-Uniform Charge Density

local or relative minimum

1 Max/Min Problems: \(2^\text{nd}\) Derivative Test

local maximum

3 Functions of Several Variables: Functions of \(1\) Variable

3 Max-Min Problems: Functions of \(2\) Variables

local or relative extrema

3 Max-Min Problems: Local Minima, Local Maxima and Inflection Points of Functions of \(1\) Variable

local minimum

3 Functions of Several Variables: Functions of \(1\) Variable

3 Max-Min Problems: Functions of \(2\) Variables

local or relative minimum

3 Max-Min Problems: Local Minima, Local Maxima and Inflection Points of Functions of \(1\) Variable

local or relative maximum

3 Max-Min Problems: Local Minima, Local Maxima and Inflection Points of Functions of \(1\) Variable

local or relative maximum

1 Max/Min Problems: \(2^\text{nd}\) Derivative Test

local or relative extrema

1 Max/Min Problems: \(2^\text{nd}\) Derivative Test

locate

3 Vectors: Vectors as Arrows

locate

1 Vectors: Vectors as Arrows

located scalar

3 Vectors: Free vs. Located

located vector

3 Vectors: Free vs. Located

logistic growth

2 Applications of Differential Equations: Logistic Growth

logistic growth model

2 Applications of Differential Equations: Logistic Growth

lower bound

2 Sequences: Bounded Sequences

2 Numerical Integration: Left, Right and Middle Riemann Sums - For a Known Antiderivative

LPMD's

3 Max-Min Problems: Second Derivative Test with More Variables (Optional)

Maclaurin Series Theorem

2 Taylor Series: Taylor Series Definition

Maclaurin series

2 Taylor Series: Taylor Series Definition

magnitude

1 Vectors: Vectors as Arrows

magnitude

3 Vectors: Vectors as Arrows

magnitude

2 Partial Fractions: Complex Methods (Optional)

mass density

3 Divergence, Curl and Potentials: The Divergence of a Gradient (\(\vec\nabla\cdot\vec\nabla f\)) - The Laplacian

mass

3 Multiple Integrals in Rectangular Coordinates: Review of Center of Mass of a Bar

3 Parametric Surfaces and Surface Integrals: Applications of Double Integrals - Center of Mass

matrix is diagonal

3 Cross Product: Properties of Determinants

Mean Value Theorem for Derivatives

2 Taylor Series: undefined

Middle Riemann Sum Error and Bound Theorem

2 Numerical Integration: Error Bounds - Trapezoid Rule and Middle Riemann Sum

Midpoint of a Line Segment

3 Coordinate Systems: Distance Formula in 2D

mixed second partials

3 Higher Order Derivatives: Higher Derivatives

Mixing problems

2 Applications of Differential Equations: Mixing Problems

Moment of Inertia of a Bar with Non-Uniform Density

2 Mass and Center of Mass: Moment of Inertia - Bar with Non-Uniform Density

moment about the \(x\)-axis

3 Multiple Integrals in Rectangular Coordinates: Applications of Double Integrals - Center of Mass

moment

3 Multiple Integrals in Rectangular Coordinates: Applications of Double Integrals - Centroid of a Region

moment of inertia

2 Mass and Center of Mass: Moment of Inertia - Bar with Non-Uniform Density

moment about the \(y\)-axis

3 Multiple Integrals in Rectangular Coordinates: Applications of Double Integrals - Center of Mass

moments of the area

3 Parametric Surfaces and Surface Integrals: Centroid

moments of the mass

3 Parametric Surfaces and Surface Integrals: Center of Mass

monkey saddle

3 Functions of Several Variables: Functions of \(2\) Variables

monotonic

2 Sequences: Monotonic Sequences

Multiple of a Row (<strong>Row Operation II</strong>)*

3 Cross Product: Properties of Determinants

Multiplication of Power Series by Polynomials

2 Power Series: Multiplication by Polynomials

multiplicative constant of integration

2 Applications of Differential Equations: Exponential Growth and Decay

multiplicity

2 Partial Fractions: General Decompositions

\(n\)D Spherical Coordinates

3 Coordinate Systems: Cylindrical and Spherical Coordinates in Higher Dimensions (Optional)

\(n\)D Cylindrical Coordinates

3 Coordinate Systems: Cylindrical and Spherical Coordinates in Higher Dimensions (Optional)

nearby

3 Functions of Several Variables: Functions of \(1\) Variable

3 Max-Min Problems: Functions of \(2\) Variables

negative infinity

1 Precise Limits and Continuity: Precise Limits at Infinity

negative

2 Convergence of General Series: Overview

2 Convergence of Positive Series:

negative of a vector \(\vec v\)

1 Dot Product: Properties

negative infinity

2 Sequences: Precise Definiton of Limits

2 Improper Integrals: Integrals Improper at an Infinite Endpoint

negative of a vector \(\vec v\)

3 Dot Product: Properties

Net Electric Charge on a Bar with Non-Uniform Charge Density

2 Mass and Center of Mass: Electric Charge - Bar with Non-Uniform Charge Density

net charge

2 Mass and Center of Mass: Electric Charge - Bar with Non-Uniform Charge Density

net electric charge

2 Mass and Center of Mass: Electric Charge - Bar with Non-Uniform Charge Density

Newton's Law of Heating and Cooling

2 Applications of Differential Equations: Newtonian Heating and Cooling

Newton's Law of Gravity

3 Divergence, Curl and Potentials: The Divergence of a Gradient (\(\vec\nabla\cdot\vec\nabla f\)) - The Laplacian

Newton's Second Law of Rotational Motion

3 Cross Product: Application - Balancing Torques

Non-Existence of a Scalar Potential

3 Divergence, Curl and Potentials: Non-Existence of a Scalar Potential

Non-Existence of a Vector Potential

3 Divergence, Curl and Potentials: Non-Existence of a Vector Potential

non-parametric equations

3 Lines and Planes: Planes - Non-Parametric Equations

Non-Parametric Equation of a Line

3 Lines and Planes: Lines - Non-Parametric Equations

non-parametric equation of the line

3 Lines and Planes: Lines - Non-Parametric Equations

Non-Uniqueness of Antiderivatives

2 Antiderivatives: Non-Uniqueness of Antiderivatives

normal equation

3 Lines and Planes: Lines in \(\mathbb{R}^2\) - More on the Normal Equation

normal form

3 Lines and Planes: Lines in \(\mathbb{R}^2\) - Non-Parametric Equations

normal acceleration

3 Properties of Curves: Tangential and Normal Acceleration

normal differential of arc length

3 Line Integrals: Normal Line Integrals in \(\mathbb{R}^2\)

normal line integral of a vector field

3 Line Integrals: Normal Line Integrals in \(\mathbb{R}^2\)

normal vector

3 Parametric Surfaces and Surface Integrals: Normal Vector

number of terms

2 Series: Geometric Series - Finite

octants

3 Coordinate Systems: 3D Rectangular Coordinates

optimize

3 Max-Min Problems: Constrained Max-Min Problems

optimized function

3 Max-Min Problems: Constrained Max-Min Problems

order

2 Differential Equations and Initial Value Problems: What is a Differential Equation?

ordered triple notation

3 Vectors: \(ijk\) Notation

ordered triple notation

1 Vectors: \(\hat\imath\hat\jmath\) Notation

orientable

3 Parametric Surfaces and Surface Integrals: Vector Surface Integrals

orientation

3 Parametric Surfaces and Surface Integrals: Vector Surface Integrals

orientation of the curve

3 Line Integrals: Flow and Circulation

oriented

3 Parametric Surfaces and Surface Integrals: Vector Surface Integrals

orienteering

3 Vectors: Application: Orienteering

origin

3 Coordinate Systems: 2D Rectangular Coordinates

original series

2 Convergence of Positive Series: Overview

oscillatory divergent

2 Sequences: Limits - Definition

osculating circle

3 Properties of Curves: Curvature

outer

3 Chain Rule: Chain Rule with \(1\) Intermediate Variable

outer

1 Chain Rule & Implicit Differentiation: Chain Rule

output tolerance

3 Limits and Continuity: Precise Limits for Functions of \(1\) Variable (Optional Review)

output tolerance

2 Sequences: Precise Definiton of Limits

output bound

1 Precise Limits and Continuity: Precise Infinite Limits

output tolerance

1 Precise Limits and Continuity: Precise Limits at Infinity

\((p,q)\)-moment

3 Multiple Integrals in Rectangular Coordinates: Applications of Double Integrals - Centroid of a Region

\(p\)-series

2 Convergence of Positive Series: The Integral Test

parabola

3 Curves and Surfaces: Quadratic Curves - Part 2

parabolic cylinders

3 Curves and Surfaces: Quadratic Surfaces - Part 2

parabolic cylinder

3 Curves and Surfaces: Quadratic Surfaces - Part 2

paraboloids

3 Curves and Surfaces: Quadratic Surfaces - Part 2

Parallel and Perpenducular Scalar Projections

3 Dot Product: Scalar Projections

Parallel Vector Projection

1 Vectors: Vector Projections

Parallel Vector Projection

3 Dot Product: Vector Projections

Parallel and Perpenducular Scalar Projections

1 Vectors: Scalar Projections

Parallelogram Method

1 Vectors: Vector Addition - Geometric

Parallelogram Method

3 Vectors: Vector Addition: Geometric Construction

parameter

3 Curves and Surfaces: Curves in \(\mathbb{R}^2\)

3 Lines and Planes: Curves in \(\mathbb{R}^3\)

parameter

1 Vectors: Parametric Curves

parameters

3 Curves and Surfaces: Surfaces in \(\mathbb{R}^3\)

3 Lines and Planes: Planes - Parametric Equations

Parametric Equation of a Line

1 Vectors: Parametric Lines

parametric equation of the plane

3 Lines and Planes: Planes - Parametric Equations

parametric curve

1 Vectors: Parametric Curves

parametric equation

3 Lines and Planes: Lines - Parametric Equations

parametric equation of the line

3 Lines and Planes: Lines - Parametric Equations

Parametric Equation of a Line

3 Lines and Planes: Lines - Parametric Equations

Parametric Equation of a Plane

3 Lines and Planes: Planes - Parametric Equations

parametric equation of the line

1 Vectors: Parametric Lines

parametric equation

1 Vectors: Parametric Lines

parametric surface

3 Curves and Surfaces: Surfaces in \(\mathbb{R}^3\)

3 Parametric Surfaces and Surface Integrals: Introduction to Parametric Surfaces

parametric curve

3 Curves and Surfaces: Curves in \(\mathbb{R}^2\)

3 Parametric Surfaces and Surface Integrals: Curves in \(\mathbb{R}^3\)

parametric curve

2 Arc Length and Surface Area: Arc Length of a Parametric Curve

Parametrization Method

3 Max-Min Problems: Multiple Constraint Problems (Optional)

parametrized region

3 Parametric Surfaces and Surface Integrals: Introduction to Parametric Surfaces

Parametrizing the Graph of a Function

2 Arc Length and Surface Area: Parametric Curves

partial derivatives

3 Partial Derivatives and Tangent Planes: Algebraic Definition of Partial Derivatives

partial

3 Partial Derivatives and Tangent Planes: Computation

partial derivative

3 Partial Derivatives and Tangent Planes: Computation

partition points

3 Multiple Integrals as Riemann Sums: Riemann Sum Definition of Single Integrals (Review)

path

2 Arc Length and Surface Area: Parametric Curves

path independent integrals

3 Fundamental Theorem of Calculus for Curves: Application - Path Independence

Perpendicular Vector Projection

1 Vectors: Vector Projections

Perpendicular Vector Projection

3 Dot Product: Vector Projections

\(\phi\)-curve

3 Coordinate Systems: 3D Spherical Coordinates

pivot point

3 Cross Product: Application - Balancing Torques

pivot point