Determine whether the following vector fields are conservative on the given domain and if so,...

Question:

Determine whether the following vector fields are conservative on the

given domain and if so, give the potential function (up to a constant) of which the field is

the gradient.

a) {eq}F(x,y,z)= \langle x^3+yz, y+xz, \frac{1}{z}+xy \rangle {/eq} on all of space except for the origin.

b) {eq}F(x,y)=(x-3x^2y, 1+xy){/eq} in the plane.

Conservative Vector Field:

For a vector field to be conservative, there exists a scalar function f called scalar potential such that F can be expressed as the gradient of f, {eq}F=\nabla f {/eq}, which implies that curl F = 0. So, if {eq}\nabla \times F\neq 0, {/eq} then it can be inferred that F is not conservative.

Answer and Explanation:

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a) For a vector field to be conservative, there exists a scalar function f such that F can be expressed as the gradient of f, {eq}\displaystyle...

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Conservative Forces: Examples & Effects

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Chapter 5 / Lesson 8
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Learn how to tell if a force is conservative and what exactly is being conserved. Then look at a couple of specific examples of forces to see how they are conservative.


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