9. Line Integrals

f. Normal Line Integrals in \(\mathbb{R}^2\) (Optional)

There is one other type of line integral which only exists in \(\mathbb{R}^2\). Recall that when we discussed the equation of a line, we said that a line can always be specified by giving a point and a tangent vector, but in \(\mathbb{R}^2\) it can also be specified by giving a point and a normal vector. If the tangent vector is \(\vec v=\langle v_1,v_2\rangle\), then the normal vector is \(\vec n=\vec v^\perp=\langle v_2,-v_1\rangle\). From these we get the unit tangent vector: \[ \hat v=\dfrac{\vec v}{|\vec v|} \qquad (=\hat T) \] and the unit normal vector: \[ \hat n=\dfrac{\vec n}{|\vec n|}=\dfrac{\vec n}{|\vec v|} \] Here, we have noted that the unit tangent vector is also called \(\hat T\) and \(|\vec n|=|\vec v|\).

In the same way, the usual line integral of a vector: \[ \int_A^B \vec F\cdot d\vec s =\int_A^B \vec F\cdot\hat v\,ds \] should be regarded as a tangential line integral since it is the scalar line integral of the tangential component of \(\vec F\): \[ F_\parallel=\text{comp}_{\vec v}\vec F =\dfrac{\vec F\cdot\vec v}{|\vec v|} =\vec F\cdot\hat v \] On the other hand, in \(\mathbb{R}^2\), the normal component of \(\vec F\) is: \[ F_\perp=\text{comp}_{\vec n}\vec F =\dfrac{\vec F\cdot\vec n}{|\vec n|} =\vec F\cdot\hat n \] So in \(\mathbb{R}^2\), we can use \(\vec n\) instead of \(\vec v\) to also define a normal line integral.

The normal differential of arc length is: \[ d\vec n=d\vec s^\perp=(dy,-dx) \] For a parametrized curve, \(\vec r(t)=(x(t),y(t))\), it can be expressed in terms of the normal, \(\vec n=\vec v^\perp=\left(\dfrac{dy}{dt},-\,\dfrac{dx}{dt}\right)\), as \[ d\vec n =\left(\dfrac{dy}{dt},-\,\dfrac{dx}{dt}\right)\,dt =\vec n\,dt \] Notice that the length of the normal differential of arc length is the same scalar differential of arclength: \[ |d\vec n|=\sqrt{dy^2+dx^2\,}=|d\vec s|=ds \]

Using the normal differential of arc length, we can now define the normal integral of a vector:

The normal line integral of a vector field \(\vec F\) along a parametric curve \(\vec r(t)=(x(t),y(t))\) between \(A=\vec r(a)\) and \(B=\vec r(b)\) is given by \[ \int_A^B \vec F\cdot d\vec n =\int_a^b \vec F(\vec r(t))\cdot\vec n\,dt \qquad (1) \] where \(\vec F(\vec r(t))\) is the composition of \(\vec F\) and \(\vec r\), i.e. the function \(\vec F(x,y)\) with \(x\) and \(y\) replaced by \(x(t)\) and \(y(t)\). The composition \(\vec F(\vec r(t))\) is called the value of the vector field along the curve.

Notice that we again switch the limits on the integral from the abstract points \(A\) and \(B\) to the values \(a\) and \(b\) of the parameter \(t\) when we switch from the integral with the general differential \(d\vec n\) to the integral with the parameter differential \(dt\).

The normal line integral can also be written in other notations.

1. Using differentials: \[ \int_A^B \vec F\cdot d\vec n =\int_A^B F_1\,dy-F_2\,dx \qquad (2) \]
2. Using the unit normal vector: \[ \int_A^B \vec F\cdot d\vec n =\int_A^B \vec F\cdot\hat{n}\,ds \qquad (3) \]

When computing a line integral of a vector field, always use the forms shown in (1) or (2) as demonstrated in the examples. Never use version (3) for computation. If you do, you end up unnecessarily computing \(|\vec v|\) since you divide by \(|\vec v|\) when forming \(\hat{n}=\dfrac{\vec n}{|\vec v|}\) and multiply by it when forming \(ds=|\vec v|\,dt\) so that they cancel out in the end. The version (3) is used mostly for theoretical purposes, since it writes the vector integral as the scalar integral of the normal component of the vector field. \[ F_\perp=\text{comp}_{\vec n}\vec F =\dfrac{\vec F\cdot\vec n}{|\vec n|} =\vec F\cdot\hat n \] This will become clear when we discuss applications on the next page.

It should also be noted that every normal line integral can be rewritten as a tangential line integral but for a rotated vector field. Specifically, if we define \(\vec F^\perp=\langle F_2,-F_1\rangle\), then \[\begin{aligned} \int_A^B &\vec F\cdot d\vec n =\int_A^B F_1\,dy-F_2\,dx \\ &=\int_A^B -F_2\,dx+F_1\,dy =-\int_A^B \vec F^\perp\cdot d\vec s \end{aligned}\] This rewrite is rarely used.

Compute \(\displaystyle \int_{(0,0)}^{(2,4)} \vec F\cdot d\vec n\) along the parabola \(\vec r(t)=\left(t,t^2\right)\) for the vector field \(\vec F=\langle 3y,2x^3\rangle\).

The value of \(\vec F\) along the curve is: \[ \vec F(\vec r(t))=\langle 3t^2,2t^3\rangle \] The velocity is \(\vec v=(1,2t)\) and the normal is \(\vec n=(2t,-1)\). So \(\vec F(\vec r(t))\cdot\vec n=6t^3-2t^3=4t^3\), and the integral is \[ \int_{(0,0)}^{(2,4)} \vec F\cdot d\vec n =\int_0^2 \vec F(\vec r(t))\cdot\vec n\,dt =\int_0^2 4t^3\,dt=\left[t^4\rule{0pt}{10pt}\right]_0^2=16 \]

We look at applications on the next page.