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For the following vector fields, determine whether or not the vector field is conservative (i.e....

Question:

For the following vector fields, determine whether or not the vector field is conservative (i.e. it can be written as a gradient of a scalar function). If it is conservative, construct a potential function and use it to evaluate the vector line integral {eq}\int\limits_C \vec F \cdot \vec d r {/eq} where {eq}C{/eq} is a path starting at {eq}(1,1) {/eq} and ending at {eq}(2,2).{/eq}
{eq}(a) \displaystyle \vec F = (1,0) .\\ (b) \displaystyle \vec F = (y, x + 1). \\(c) \displaystyle \vec F = (\frac{1}{y} - 2 x , y - \frac{x}{y^2}). {/eq}

Conservative Vector Field:

Given a vector field f, if there exists a function f(x,y) such that its gradient is equal to F, then F is conservative and f is called scalar potential. Moreover, if F is conservative, then {eq}\int_{C}F \cdot dr {/eq} is independent of the path of integration.

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If there exists a function f(x,y) such that its gradient is equal to F, then F is conservative and f is called scalar potential. Also, if...

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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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