Chapter 14: Q 32. (page 1154)

Use Green’s Theorem to evaluate the integral for the given vector field and curve.
$F (x, y) = (y^{2} + 1) i + 2 x y j$ and C is the circle with equation $x^{2} + y^{2} = 9$ transversed counterclockwise.

Short Answer

Expert verified

$\int_{c} F (x, y) d x = 0$

Step by step solution

Given Information

The given expression is $F (x, y) = (y^{2} + 1) i + 2 x y j$

Simplification

$F (x, y) = (y^{2} + 1) i + 2 x y j$

$\int_{c} F (x, y) d x = \int \int_{R} (\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) d A$

From the given equation of circle, the limits come out to be $(0, 3), (0, 3)$

Equation becomes $\int_{c} F (x, y) d x = \int_{0}^{3} \int_{0}^{3} (\frac{\partial (2 x y)}{\partial x} - \frac{\partial (y^{2} + 1)}{\partial y}) d y d x$

$\Rightarrow \int_{c} F (x, y) d x = \int_{0}^{3} [2 y - 2 y] d x$

$\Rightarrow \int_{c} F (x, y) d x = 0$

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Use Green’s Theorem to evaluate the integral for the given vector field and curve.
$F (x, y) = (y^{2} + 1) i + 2 x y j$ and C is the circle with equation $x^{2} + y^{2} = 9$ transversed counterclockwise.

Short Answer

Step by step solution

Given Information

Simplification

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Use Green’s Theorem to evaluate the integral for the given vector field and curve.Fx,y=y2+1i+2xyjand C is the circle with equation x2+y2=9 transversed counterclockwise.

Short Answer

Step by step solution

Given Information

Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Applied Mathematics

Decision Maths

Probability and Statistics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Use Green’s Theorem to evaluate the integral for the given vector field and curve.
$F (x, y) = (y^{2} + 1) i + 2 x y j$ and C is the circle with equation $x^{2} + y^{2} = 9$ transversed counterclockwise.