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# MYMathApps Calculus 3: Multi-Variable Calculus

A Web Based, Interactive, Multimedia Calculus

Doing, Not Just Viewing

Learn Multi-Variable Calculus intuitively with the interactive MYMathApps Calculus 3 online textbook

## Benefits of MYMACalc 3 over Other Texts

Many students have a great deal of difficulty visualizing in three dimensions. 3D graphics, both static and animated, are designed to help with this difficulty. Animations are also helpful with visualizing other concepts.

• In discussing Cross Products, there are animations to help students visualize the right hand rule.
• In discussing Vector Addition and Magnitude, there are animations to help students visualize the Triangle Inequality.
• Lots of graphics help students understand the derivations of Parametric and Non-Parametric equations for lines and planes, how lines in $\mathbb{R}^3$ can be skew rather than parallel or intersecting, and how to compute the distance from a point to a line or plane.
• There are plots of examples of all the quadratic curves (conic sections) and all the quardatic surfaces.
• In discussing the Properties of Curves, there are animations understand arclength, the tangent, normal and binormal unit vectors, curvature and torsion.
• There is a highly interactive graphic to motivate the definition of Partial Derivatives and Tangent Planes. After viewing the graph of a function of 2 variables, a student clicks on buttons and moves sliders to construct the two traces and position them, add tangent lines to the traces and move them along the traces and finally add the tangent plane. All of this can be animated or rotated with the mouse.
• Interactive graphic help students visualize the Divergence of a Vector Field as the amount a fluid flows out of a circle and the Curl of a Vector Field as the amount a fluid rotates around a circle.
• Animations show students the convergence of Rieman Sums for 1 and 2 variables.
• When studying Green's, Stokes' and Gauss' Theorems, there are animations to help students understand the orientations of curves and surfaces and the relation between them.

MYMACalc 3 covers lots of material in a different order or with different emphasis than many standard books:

• MYMACalc 3 has students compute partial derivatives, then turns to the interpretation as the slopes of the traces and finally the limit definitions.
• MYMACalc 3 gets students to compute iterated integrals, then turns to the Riemann sum definition, where students see iterated integrals arising as the result in various coordinate systems.
• MYMACalc 3 defines derivatives with respect an arbitrary vector, not just unit vectors, because these are important when computing the derivative along a curve. Directional derivatives are then a special case.
• MYMACalc 3 emphasizes the passive view of curvilinear coordinates as a different description of the points in the plane, rather than the active view as a transformation between two planes.
• Standard books cover line and surface integrals with the theorems, after partial derivatives and multiple integrals. They frequently spend time on line and surface integrals for graphs and then repeat for parametric surfaces.
• MYMACalc 3 treats line and surface integrals almost exclusively parametrically, with graphs of functions as special cases.
• MYMACalc 3 covers line integrals with arc length integrals. This gives students time to digest line integrals before they get to surface integrals or the theorems.
• MYMACalc 3 covers surface integrals directly after multiple integrals emphasizing the analogy between curvilinear coordinates in the plane and parametric surfaces. This is separated from Stokes' and Gauss' Theorems by the Fundamental Theorem of Calculus for Curves and Green's Theorem.
• MYMACalc 3 treats the four Fundamental Theorems of Vector Calculus equally, emphasizing that each theorem uses the Fundamental Theorem of Calculus to reduce the number of integrals by one and replace the domain of integration by the boundary of the original domain.

## Benefits of an Online Text

Although primarily a standard calculus text, the online format allows for many features which are impossible in a static book:

• There are many animations directly in the pages, some of which can be modified real-time by the student.
• The text is fully hyperlinked. The material is designed to be followed linearly, with frequent side excursions for extra material such as additional examples, background information, proofs or advanced material.
• Most pages in the text cover a single topic followed by an example and an exercise. More exercises appear on a linked exercise page. Hints, answers and full solutions for the exercises toggle in dropdown regions directly below the exercise. Hopefully, students try the problems before they click on the answer or solution button. The solution is not right there in front of them as in a paper book.
• When there is an obvious way to check an answer, the check appears in a light green region at the end of the an example's solution, or in a separate drop down region after an exercise. For example:
After solving an equation, one should check it by substituting the solution back into the equation.
After finding an antiderivative, one should check by differentiating.
After finding the center of mass of an object, one should check that it is inside (the convex hull of) the object.
• Sometimes there is a comment or remark about an exercise, that the author would like to make which might have given away the answer if visible in the page. In that case there is an additional button at the end of the exercise to drop down the remark. For example:
The integral you just computed gives the area of a semi-circle. So you could have known the answer in advance.
• Many books give formulas without proof. MYMACalc proves every formula. Derivations and proofs are directly in the text when they are essential to understanding, but are frequently hidden in a drop-down region when they are not essential, so they do not obstruct the flow of the material but are still easily accessible to interested students. There are many more proofs than most textbooks.
• There are exercise pages for each chapter and the exercises are grouped by section of the chapter. Each group is hyperlinked back to the appropriate pages, to facilitate those students who like to start on the exercises and only read the chapter when they need help doing the exercises.
• There are links to the Maplets for Calculus (M4C) directly in the text. These drill the students on specific topics acting like a “Tutor without the Tutor”. Since the M4C are displayed in Java, they cannot be viewed on most mobile devices. Consequently, they are gradually being converted into browser-based tutorials which either are directly embedded in the text or appear on separate tutorial pages.
• MYMACalc works in any browser on any device, except for the Maplets for Calculus (M4C) (because they are Java based).