Stoke’s Theorem

Section 5.9 Stoke’s Theorem

Subsection 5.9.1 Statement

The following is an integral identity that is useful for some calculations.

Theorem 5.9.1. Stoke’s.

For a oriented, smooth surface \(D\) with positive orientation inside a simple, closed curve \(C\) and a vector field \(\vec{F}(x,y,z)\) whose derivatives are continuous in the surface

\begin{align} \oint_C \vec{F} \cdot d\vec{r} = & \int_a^b \vec{F}(x(t),y(t),z(t)) \cdot \left( \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right) \; dt\notag\\ = & \iint_D \mbox{curl } \vec{F} \cdot d\vec{S}\notag\\ = & \iint_D \mbox{curl } \vec{F} \cdot \hat{n} \; dS\notag\\ = & \iint_D \mbox{curl } \vec{F} \cdot (\vec{r}_u \times \vec{r}_v) \; dA\tag{5.2} \end{align}

The version of the right hand side in (5.2) indicates that in practice when we can parameterize the surface, we do not need to calculate a normalized version of the curve. The reason is the same as for the divergence theorem (note the change from \(dS\) to \(dA\)).

Activity 79.

(a)

Compare Green’s and Stoke’s Theorems below.

\begin{align*} \oint_C \vec{F} \cdot d\vec{r} = & \displaystyle \dint_D \left( \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) \; dx \: dy. & \text{Green's Theorem}\\ \oint_C \vec{F} \cdot d\vec{r} = & \displaystyle \dint_D \mbox{curl } \vec{F} \cdot \vec{n} \; dS. & \text{Stoke's Theorem} \end{align*}

Exercises 5.9.2 Exercises

1.

Calculate the effect of the vector field \(F(x,y,z)=\langle yz,xz,0 \rangle\) on the outer edge of the spiral ramp \(S(r,\theta)=(r\cos\theta,r\sin\theta,\theta)\) with \(r \in [0,10]\) and \(\theta \in [0,2\pi].\) The parameterized path is \(\vec{r}(\theta)=(10\cos\theta,10\sin\theta,\theta).\)

2.

Use Stoke’s Theorem to calculate the effect.

Lessons for Multivariable Calculus

Search Results:

Section 5.9 Stoke’s Theorem

Subsection 5.9.1 Statement

Theorem 5.9.1. Stoke’s.

Activity 79.

(a)

Exercises 5.9.2 Exercises

1.

2.