Verify that the line integral and the surface integral of...

Verify that the line integral and the surface integral of Stokes' Theorem are equal for the following vector fields, surfaces $S,$ and closed curves $C .$ Assume that C has counter-clockwise orientation and S has a consistent orientation. $\mathbf{F}=\langle y-z, z-x, x-y\rangle ; S$ is the cap of the sphere $x^{2}+y^{2}+z^{2}=16$ above the plane $z=\sqrt{7}$ and $C$ is the boundary of $S$

Verify that the line integral and the surface integral of Stokes' Theorem are equal for the following vector fields, surfaces $S,$ and closed curves $C .$ Assume that C has counter-clockwise orientation and S has a consistent orientation.
$\mathbf{F}=\langle y-z, z-x, x-y\rangle ; S$ is the cap of the sphere $x^{2}+y^{2}+z^{2}=16$ above the plane $z=\sqrt{7}$ and $C$ is the boundary of $S$

Key Concepts

Stokes' Theorem

Stokes' Theorem is a fundamental result in vector calculus that relates a surface integral over a surface S to a line integral around its boundary curve C. Specifically, it states that the integral of the curl of a vector field over the surface is equal to the circulation of the vector field along the boundary of that surface. This theorem provides a powerful tool for converting difficult surface integrals into simpler line integrals and vice versa.

Curl of a Vector Field

The curl of a vector field quantifies the rotation or swirling strength at a given point in the field. It is defined as the cross-product of the del operator with the given vector field. In the context of Stokes' Theorem, the curl appears in the surface integral, and its computation is essential for evaluating how the field circulates around infinitesimal loops on the surface.

Line Integral

A line integral of a vector field over a curve computes the accumulation of the field's effect along that curve. When applied to closed curves, such integrals measure the total circulation of the field around the boundary. This concept is integral to applying Stokes' Theorem, which equates this circulation with the flux of the curl through the surface bounded by the curve.

Surface Integral

The surface integral of a vector field over a surface involves summing contributions from the field over each infinitesimal area element on the surface. For Stokes' Theorem, the surface integral is taken over the curl of the vector field, capturing the net rotation over the entire surface, which corresponds to the total circulation along the boundary.

Orientation

Orientation is a crucial aspect in applying Stokes' Theorem. It refers to the consistent choice of direction for traversing the boundary curve and the corresponding direction of the normal vector on the surface. The correct alignment (often ensured by the right-hand rule) is essential to ensure that the line and surface integrals are computed consistently and the theorem holds true.

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