7-10 Use Stokes' Theorem to evaluate ∫C 𝐅 ·d 𝐫 . In each case C is oriented counterclockwise as

step 1 We are given a vector field F and a curve C oriented counterclockwise as viewed from above. We a

7-10 Use Stokes' Theorem to evaluate ∫C 𝐅 ·d 𝐫 . In each...

$7-10$ Use Stokes' Theorem to evaluate $\int_{C} \mathbf{F} \cdot d \mathbf{r} .$ In each case $C$ is oriented counterclockwise as viewed from above. $$\begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+2 z \mathbf{j}+3 y \mathbf{k}, \quad C \text { is the curve of intersec- }} \\ {\text { tion of the plane } x+z=5 \text { and the cylinder } x^{2}+y^{2}=9}\end{array}$$

$7-10$ Use Stokes' Theorem to evaluate $\int_{C} \mathbf{F} \cdot d \mathbf{r} .$ In each case $C$ is
oriented counterclockwise as viewed from above.
$$\begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+2 z \mathbf{j}+3 y \mathbf{k}, \quad C \text { is the curve of intersec- }} \\ {\text { tion of the plane } x+z=5 \text { and the cylinder } x^{2}+y^{2}=9}\end{array}$$

Key Concepts

Stokes' Theorem

Stokes' Theorem is a fundamental result in vector calculus that relates the line integral of a vector field around a closed curve to the surface integral of the curl of that vector field over a surface bounded by the curve. It effectively transforms a potentially difficult line integral into a usually simpler surface integral, provided the right surface is chosen with the correct orientation.

Curl of a Vector Field

The curl of a vector field is a vector operator that describes the infinitesimal rotation of the field at a point. It provides a measure of the swirling strength and direction and is essential in applying Stokes' Theorem because the surface integral in the theorem involves the dot product of the curl with the surface's unit normal vector.

Surface Integral

A surface integral generalizes the concept of integration over a curve to integration over a two-dimensional surface. In the context of Stokes' Theorem, the surface integral evaluates the flux of the curl of a vector field across a chosen oriented surface, which is equal to the circulation of the field along the boundary of the surface.

Orientation of Surfaces and Curves

Orientation is a crucial concept when applying Stokes' Theorem, as it ensures that the integration is performed consistently. The direction of the unit normal vector on the surface must be chosen so that, when using the right-hand rule, the resulting boundary curve is traversed in the specified direction. Proper alignment of the surface's orientation with the curve's prescribed orientation is essential for correctly applying the theorem.

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