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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

https://brainmass.com/math/calculus-and-analysis/vector-calculus-applications-534999

## 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:

We have:

where:

Now the above integral is path-independent if and only if i.e., if and only if:

We have:

whence the integral is path-independent:

We wish to evaluate I over the path given by:

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:

Along this path, we have:

(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:
(ii)
We have:

(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:

We wish to compute the Jacobian of this transformation:

We have:

(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.

We have:

(e) Repeat of (c).

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

and to compute its area, A.
In polar coordinates, we have:

whence:

and hence:

A plot of C is shown below for:

We have:

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

for:

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:

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

We have: