Question

Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = (x + y2)i + (y + z2)j + (z + x2)k, C is the triangle with vertices (3, 0, 0), (0, 3, 0), and (0, 0, 3).

Answer 1

To evaluate the line integral using Stokes' Theorem, find the curl of the vector field and the area of the surface enclosed by the triangle. Substitute these values into the Stokes' Theorem equation to calculate the line integral.

Step-by-step explanation:

To evaluate the line integral ∫ F · dr using Stokes' Theorem, we need to find the curl of the vector field F(x, y, z) = (x + y^2)i + (y + z^2)j + (z + x^2)k. The curl of F is given by ∇ × F. Since the surface enclosed by the triangle C is a flat plane, the flux through the surface can be written as ∫∫ (∇ × F) · dS, where dS is the area element of the surface.

To find the curl of F, we take the partial derivatives of the components of F with respect to x, y, and z, and then calculate the determinant of the resulting matrix. Using the given values, we can then evaluate the line integral ∫ F · dr by substituting the curl of F and the area of the surface into the Stokes' Theorem equation.

Answer 2

The student's question involves applying Stokes' Theorem to find the work done by a vector field along a closed curve. The answer would entail calculating the curl of the vector field and evaluating a surface integral over the surface bounded by the curve.

Step-by-step explanation:

The problem posed is a classic example of an application of Stokes' Theorem, which is a fundamental theorem in vector calculus. To solve the given problem, we would first determine the curl of the vector field F and then apply Stokes' Theorem to transform the line integral over the closed curve C into a surface integral over the surface enclosed by C. Then, we would evaluate the surface integral over the triangle formed by the vertices (3, 0, 0), (0, 3, 0), and (0, 0, 3) to find the desired work done by the force field along the curve C.

Although the provided snippets from earlier solutions do not directly relate to the computation of the line integral using Stokes' Theorem, they do provide insight into vector operations that are necessary for solving such problems, such as dot products, cross products, and the geometric interpretation of vectors.