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    Vector Calculas and Applications

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    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    ** Please see the attached file for the complete solution **

    We wish to determine whether the following integral is path-dependent:
    I = f_c - 2ycos2xdx - sin2xdy

    In the practice problems, you must:
    - Determine if statement is correct
    - Calculate the Jacobian of transformation
    - Evaluate triple integrals

    © BrainMass Inc. brainmass.com December 24, 2021, 11:05 pm ad1c9bdddf
    https://brainmass.com/math/calculus-and-analysis/vector-calculus-applications-534999

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    SOLUTION This solution is FREE courtesy of BrainMass!

    ** Please see the attached file for the complete solution to the above problems **
    Solutions are found by determining if integrals are path-independent and computing given vector fields.
    (a) We wish to determine whether the following integral is path-independent:
    (please see the attached file)

    We have:
    (please see the attached file)

    where:
    (please see the attached file)

    Now the above integral is path-independent if and only if i.e., if and only if:
    (please see the attached file)

    We have:
    (please see the attached file)

    whence the integral is path-independent:
    (please see the attached file)

    We wish to evaluate I over the path given by:
    (please see the attached file)

    Since I is path-independent, we may choose any path going from the endpoint (please see the attached file) to the endpoint (please see the attached file). The simplest such path is the straight-line path, given by:
    (please see the attached file)

    Along this path, we have:
    (please see the attached file)

    (b) We are given the vector function and the scalar function We wish to verify the following formulas:

    (i) (please see the attached file)

    We have:
    (please see the attached file)
    (ii)
    (please see the attached file)
    We have:
    (please see the attached file)

    (c) We are given the following transformation from Cartesian coordinates (please see the attached file) with x and y positive to coordinates (please see the attached file) given by:
    (please see the attached file)

    We wish to compute the Jacobian of this transformation:
    (please see the attached file)

    We have:
    (please see the attached file)

    (d) We are given the vector field (please see the attached file) as well as the curve C parameterized by (please see the attached file) for (please see the attached file) We wish to compute the line integral (please see the attached file) of F along C.
    (please see the attached file)

    We have:
    (please see the attached file)

    (e) Repeat of (c).

    (f) We wish to sketch the domain D enclosed by the curve C given by:
    (please see the attached file)

    and to compute its area, A.
    (please see the attached file)
    In polar coordinates, we have:
    (please see the attached file)

    whence:
    (please see the attached file)

    and hence:
    (please see the attached file)

    A plot of C is shown below for:
    (please see the attached file)

    We have:
    (please see the attached file)

    (g) We are given the following coordinate transformation from Cartesian to spherical coordinates:
    (please see the attached file)

    for:
    (please see the attached file)

    Geometrically, r is the distance of a given point from the origin, (please see the attached file) is the azimuthal angle of P, which is the angle the ray OQ makes with the x-axis, where Q is the projection of P onto the xy-plane, and (please see the attached file) is the polar angle of P, which is the angle the ray OP makes with the z-axis.

    The Jacobian (please see the attached file) is given by:
    (please see the attached file)

    (h) We wish to evaluate the following triple integral and to construct its domain of integration:
    (please see the attached file)

    We have:
    (please see the attached file)

    The domain of integration is the region D defined by:
    (please see the attached file)

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 11:05 pm ad1c9bdddf>
    https://brainmass.com/math/calculus-and-analysis/vector-calculus-applications-534999

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