Chapter 6: Q29MP (page 338)

$∮ V \times dr$ around the boundary of the square with vertices $(1, 0), (0, 1), (- 1, 0), (0, - 1)$ , if $V = x^{2} i + 5 xj$

Short Answer

Expert verified

The solution is $\int V \times dr = 10$

Step by step solution

Given Information.

The given information is, $V = x^{2} i + 5 xj$

Definition of Green’s Theorem.

Green's theorem connects a line integral around a simple closed curve C to a double integral over the plane region D circumscribed by C in vector calculus. Stokes' theorem has a two-dimensional special case.

Find the solution.

For, $V = x^{2} i + 5 xj$

$\int V \times dx = \int x^{2} dx + 5 xdy$

Use Green’s theorem $\iint_{A} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dxdy = ∮_{\partial A} Pdx + Qdy$ , where $\partial A$ is the boundary of the area $A$ .

It can be seen that

$P = x^{2} \to \frac{\partial P}{\partial y}$

$= 0$

role="math" $Q = 5 x \to \frac{\partial Q}{\partial x}$

$= 5$

Since the contour is the boundary of the square with the given vertices, the area $A$ is then the area of the square.

$\int V \times dr = 5 {(\sqrt{(1 - 0)^{2} + (0 - 1)^{2}})}^{2}$

$= 10$

Hence, the solution is $\int V \times dr = 10$ .

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$∮ V \times dr$ around the boundary of the square with vertices $(1, 0), (0, 1), (- 1, 0), (0, - 1)$ , if $V = x^{2} i + 5 xj$

Short Answer

Step by step solution

Given Information.

Definition of Green’s Theorem.

Find the solution.

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∮V×draround the boundary of the square with vertices(1,0),(0,1),(−1,0),(0,−1) , ifV=x2i+5xj

Short Answer

Step by step solution

Given Information.

Definition of Green’s Theorem.

Find the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Astrophysics

Oscillations

Space Physics

Atoms and Radioactivity

Turning Points in Physics

Nuclear Physics

Study anywhere. Anytime. Across all devices.

$∮ V \times dr$ around the boundary of the square with vertices $(1, 0), (0, 1), (- 1, 0), (0, - 1)$ , if $V = x^{2} i + 5 xj$