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    Solve: Multivariate Calculus

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    Please see the attached file for full problem description.

    (a) where R is the region in the first quadrant which lies inside the circle x^2 + y^2 = 2x and outside the circle x^2 + y^2 = 1.

    (b) If f(x,y) = x/y and h(x,y) = (1/2)x^2 - 4y find the rate of change of f(x,y) at the point (2, -1) in the direction in which h(x,y) is increasing most rapidly.

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    Solution Preview

    (a) Whenever you see a double integral involving (x^2 + y^2) try transforming using polar coordinates to see if it simplifies. Remember the relations for polar transformations:

    x = r cos theta y=rsin theta
    x^2+y^2 = r^2
    and the Jacobian gives
    dxdy = r dr dtheta

    Now the region of interest R is the segment in the 1st quadrant bounded by the circles with radii 1 and sqrt(2*pi). In polar coordinates since we are restricted to the first quadrant we have 0 < theta < pi/2. For the r variable we must restrict ...

    Solution Summary

    This solution provides a very detailed, step by step response demonstrating how to approach a problem with a double integral, and another problem dealing with the rate of change. An explanation accompanies each step. A pdf. file is attached which contains the complete solution.