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Vector Fields, Fundamental Theorem of Line Integrals

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1. Find the curl of the vector field F at the indicated point:

2. Evaluate the following line integral using the Fundamental theorem of line Integrals:

3. Use Green's Theorem to calculate the work done by the force F in moving a particle around the closed path C:

4. Find the area of the surface over the part of the plane:

5. Use the Divergence Theorem to evaluate and find the outward flux of F through the surface of the solid bounded by the graphs of the equations:

6. Verify Stoke's Theorem by evaluating as a line integral:

See attached file for full problem description.

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Vector Fields, Fundamental Theorem of Line Integrals, Green's Theorem, Divergence Theorem and Stokes' Theorem are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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1. Find the curl of the vector field F at the indicated point:

The curl of the vector field is

So at (2, -1, 3) the curl is

2. Evaluate the following line integral using the Fundamental theorem of line Integrals:

Suppose that C is a smooth curve given by , . Also suppose that f is a function whose gradient vector, , is continuous on C. Then,

So first we need to find the function whose gradient vector is

Now we differentiate f(x, y) respect to y:

Where c is constant.
So
Thus, the line integral is

3. Use Green's Theorem to calculate the work done by the ...

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