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Vector Calculas and the Applications

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1. (i) Find the total derivative and the Jacobian for the following change of variables:

x = acos(uv)
y = bsin(vw)
z = xexp(-uw)

(ii) Simplify the equation: see attached
using the change of variables: see attached

See attached.

2. Find the Jacobian Jpar of the coordinates transformation for the parabolic coordinates:

See attached.

Draw the (schematic) picture of the grid for the parabolic coordinates (that is, the set of lines of constant values for u and v

3. Evaluate the following integrals:

(i) see attached

(ii) see attached

In each case sketch the domain of integration!

4. What would be the best change of variables in the double integral which yields the simplest
form?

See attached.

Write the integral in the new variables and sketch the domain of integration for the new variables. Assume that f(x) is even, i.e. that f(-t)=f(t)

5. Compute the double integral of the function (see attached) over the upper semi-circle of radius r=1. Which coordinates are more suitable for this computation?

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