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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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    https://brainmass.com/math/integrals/vector-fields-fundamental-theorem-line-integrals-118615

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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 ...

    Solution Summary

    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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