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Chapter 13: Problem 23

State Stokes's Theorem.

Short Answer

Expert verified
Stokes' Theorem states that the integral of the curl of a vector field over a surface is equal to the line integral of the vector field along the boundary of the surface. Mathematical expression: \( \int_{S} (\nabla \times \vec{F})\.d\vec{S} = \oint_{\partial S} \vec{F} \cdot d\vec{r} \)

Step by step solution

01

Introduction of Stokes' Theorem

Stokes' Theorem is a statement about the integration of differential forms. In simple terms, it's a method to transform a surface integral of a vector field into a line integral on the boundary of the surface.
02

Statement of Stokes' Theorem

The formal statement of Stokes' Theorem in its most basic form is: The surface integral of the curl of a vector field over a surface S is equal to the line integral of the vector field over the boundary of S. Mathematically, this is written as: \[ \int_{S} (\nabla \times \vec{F})\.d\vec{S} = \oint_{\partial S} \vec{F} \cdot d\vec{r} \] Where: \( \vec{F} \) is a vector field, \( S \) is a surface, \( \partial S \) is the boundary of \( S \), and \( \nabla \times \vec{F} \) is the curl of \( \vec{F} \).
03

Explanation of the terms in the formula

In this formula, on the left side we have a surface integral of the curl of a vector field, which could represent the rotation of a vector field. The right side consists of a line integral that captures the magnitude and direction of the field along the boundary of the surface.

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Most popular questions from this chapter

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y)=3 x \mathbf{i}+4 y \mathbf{j}\) \(\quad C: \mathbf{r}(t)=2 \cos t \mathbf{i}+2 \sin t \mathbf{j}, \quad 0 \leq t \leq \pi / 2\)

Demonstrate the property that \(\int_{C} \mathbf{F} \cdot d \mathbf{r}=\mathbf{0}\) regardless of the initial and terminal points of \(C,\) if the tangent vector \(\mathbf{r}^{\prime}(t)\) is orthogonal to the force field \(\mathbf{F}\) \(\mathbf{F}(x, y)=-3 y \mathbf{i}+x \mathbf{j}\) \(C: \mathbf{r}(t)=t \mathbf{i}-t^{3} \mathbf{j}\)

Find the divergence of the vector field \(\mathbf{F}\) at the given point. $$ \begin{array}{lll} \text { Vector Field } & & \text { Point } \\ \mathbf{F}(x, y, z)=x^{2} z \mathbf{i}-2 x z \mathbf{j}+y z \mathbf{k} & & (2,-1,3) \\ \end{array} $$

Evaluate \(\int_{C}\left(x^{2}+y^{2}\right) d s\) \(C:\) counterclockwise around the circle \(x^{2}+y^{2}=1\) from (1,0) to (0,1)

Find \(\operatorname{curl}(\operatorname{curl} \mathbf{F})=\nabla \times(\nabla \times \mathbf{F})\). \(\mathbf{F}(x, y, z)=x^{2} z \mathbf{i}-2 x z \mathbf{j}+y z \mathbf{k}\)
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