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In order to use Green's Theorem, certain conditions must be met. (a) Which of the following...

Question:

In order to use Green's Theorem, certain conditions must be met.

(a) Which of the following conditions must be true? Select all that apply.

  • 1. {eq}\, \vec{F} \, = \, \nabla f {/eq}
  • 2. {eq}\, \vec{F} \, {/eq} must be in the direction of {eq}\, C {/eq}.
  • 3. {eq}\, R \, {/eq} is on the left as we move around the curve {eq}\, C {/eq}.
  • 4. {eq}\, R \, {/eq} is on the right as we move around the curve {eq}\, C {/eq}.
  • 5. {eq}\, \vec{F} \, {/eq} is a smooth field on an open region containing {eq}\, C \, {/eq} and {eq}\, R {/eq}.
  • 6. {eq}\, C \, {/eq} is a piecewise smooth simple closed curve that is the boundary of {eq}\, R {/eq}.

(b) If all of the conditions are met, how do you evaluate {eq}\, \displaystyle{\int_C \, \vec{F} \, \times \, d \vec{r}} {/eq}?

  • 1. {eq}\, \displaystyle{\int_R \, \left ( \frac{\partial F_2}{\partial y} \, - \, \frac{\partial F_1}{\partial x} \right ) \, dA} {/eq}
  • 2. {eq}\, \displaystyle{\int_R \, \left ( \frac{\partial F_1}{\partial x} \, - \, \frac{\partial F_2}{\partial y} \right ) \, dA} {/eq}
  • 3. {eq}\, \displaystyle{\int_R \, \left ( \frac{\partial F_1}{\partial y} \, - \, \frac{\partial F_2}{\partial x} \right ) \, dA} {/eq}
  • 4. {eq}\, \displaystyle{\int_R \, \left ( \frac{\partial F_2}{\partial x} \, - \, \frac{\partial F_1}{\partial y} \right ) \, dA} {/eq}

Green's Theorem:

The theorem which relates the line integral with the surface integral is called Green's Theorem {eq}\oint F_{1}dx+F_{2}dy=\int \int \left ( \frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y} \right )dxdy {/eq}

Answer and Explanation:

To solve the problem we will use Green's Theorem:

{eq}\oint F_{1}dx+F_{2}dy=\int \int \left ( \frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y} \right )dxdy {/eq}

here we will find the partial derivatives of the function and also to solve we should have curve C which is a piecewise smooth simple closed curve that is the boundary of R.


Learn more about this topic:

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The Fundamental Theorem of Calculus

from Math 104: Calculus

Chapter 12 / Lesson 10
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