(a) Find a function f such that F = ? f and (b) Use (a) to evaluate ? C F ? d r along the...

Question:

(a) Find a function {eq}f {/eq} such that {eq}\mathbf{F} = \nabla f {/eq} and

(b) Use (a) to evaluate {eq}\int_{C} \mathbf{F} \cdot d \mathbf{r} {/eq} along the given curve {eq}C {/eq}.

{eq}\mathbf{F}(x,y) = \left \langle(1 + xy) e^{xy}, \ x^2 e^{xy} \right \rangle {/eq}

{eq}C {/eq} is given by the vector function {eq}\mathbf{r}(t): {/eq}

{eq}\mathbf{r}(t) = \left \langle \cos t, \ 2 \sin t \right \rangle, \ 0 \leq t \leq \frac{\pi}{2} {/eq}

Fundamental Theorem of Line Integrals:

The fundamental theorem of line integrals is one of the several fundamental theorems of vector calculus.

It relates the line integral of a gradient vector field to the values of the potential function at the endpoints of the path.

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a) To find the function {eq}f {/eq} satisfying {eq}\mathbf{F} = \nabla f {/eq}, i.e., the potential function {eq}f {/eq} of {eq}\mathbf{F}(x,y) =...

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Line Integrals: How to Integrate Functions Over Paths

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Chapter 15 / Lesson 2
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Line integrals are any integral of a function that can be defined along a given curve in a three-dimensional space. Learn the process of line integration and how they can be used to map paths using parametrizations.


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