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Vector Calculas and Applications

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** 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

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Solution Summary

This solution explain how to solve various problems in multivariable calculus.

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** 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 ...

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