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Chapter 15: Problem 13

True-False Determine whether the statement is true or false. Explain your answer. Stokes' Theorem equates a line integral and a surface integral.

Short Answer

Expert verified
True, Stokes' Theorem equates a line integral to a surface integral.

Step by step solution

01

Evaluate Stokes' Theorem

Stokes' Theorem states that the line integral of a vector field \( extbf{F} \) around a closed curve \( C \) is equal to the surface integral of the curl of \( extbf{F} \) over the surface \( S \) bounded by \( C \). Mathematically, it is expressed as: \[ \int_C extbf{F} \, \cdot\, d extbf{r} = \int_S (abla \times \textbf{F}) \, \cdot\, d extbf{S} \]This equation shows the equivalence between the line integral along the curve and the surface integral over the surface.
02

Define Line and Surface Integrals

A line integral, \( \int_C extbf{F} \, \cdot\, d extbf{r} \), measures the flow of the vector field along a path or closed curve \( C \). A surface integral, \( \int_S (abla \times \textbf{F}) \, \cdot\, d extbf{S} \), measures the total curl or rotation of the vector field across a surface \( S \). Stokes' Theorem provides a way to relate these two integrals when \( C \) is the boundary of \( S \).
03

Conclude Statement Correctness

Given that Stokes' Theorem directly equates the line integral around a closed path to a surface integral over the surface bounded by that path, the statement 'Stokes' Theorem equates a line integral and a surface integral' is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Line Integral
In vector calculus, a line integral is used to calculate the work done by a vector field along a curve. Imagine you're dragging an object along a path, and you want to know how much effort it takes. This is essentially what a line integral measures.
  • A line integral deals with integrating a vector field along a specific curve.
  • The vector field, denoted as \( \mathbf{F} \), represents some force, flow, or field.
  • The notation for the line integral is \( \int_C \mathbf{F} \cdot d\mathbf{r} \), where \( C \) is the curve.
By using the line integral, we sum up the tiny bits of 'work' done along the path. This involves dot products which consider both the direction of the field and the direction of the path.
For a closed curve, the line integral evaluates properties around a loop, making it fundamental in many physical applications like electromagnetism.
Surface Integral
A surface integral extends the concept of a line integral to two dimensions, and it allows us to consider how a field interacts with a surface. Imagine a surface in space being swept by a wind. The surface integral gauges the total impact of the wind on that surface.
  • Surface integrals are used to measure a field over a given surface instead of a line.
  • The notation \( \int_S (abla \times \mathbf{F}) \cdot d\mathbf{S} \) describes how the field curls across the surface \( S \).
  • The surface is partitioned into tiny patches, and the interaction of the field with each patch is calculated.
In terms of Stokes' Theorem, the surface integral is crucial as it helps illustrate rotation or curl present across a surface. It is used to understand dynamic phenomena like fluid flow across surfaces or magnetic fields around electric currents.
Vector Field
A vector field is like an invisible force map over space. It assigns a vector to every point in a region. Examples include gravitational fields pulling objects towards Earth or sea currents showing water flow directions.
  • Each vector represents a quantity that has both magnitude and direction.
  • It can be denoted as \( \mathbf{F}(x, y, z) \) in three dimensions.
  • Vector fields are described using functions of space coordinates.
Understanding vector fields is essential because they represent the forces or flows that line and surface integrals are concerned with. These fields are foundational in physics, engineering, and various fields of applied mathematics.
Curl of a Vector Field
The curl of a vector field measures how much and in what direction the field 'twists' or 'rotates' around a point. It's like watching a whirlpool: the curl points out the axis and strength of rotation.
  • The curl is a vector that describes the infinitesimal rotation at any point in the field.
  • Mathematically, it is the result of the operation \( abla \times \mathbf{F} \).
  • Curl is used to determine the rotational characteristics of a vector field.
In the context of Stokes' Theorem, the curl relates directly to how the field behaves across the surface, which is why the surface integral involves \( abla \times \mathbf{F} \). This concept is fundamental in fluid dynamics, electromagnetism, and other physical sciences where rotational forces are present.

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Most popular questions from this chapter

A farmer weighing \(150 \mathrm{lb}\) carries a sack of grain weighing 20 lb up a circular helical staircase around a silo of radius \(25 \mathrm{ft}\). As the farmer climbs, grain leaks from the sack at a rate of \(1 \mathrm{lb}\) per \(10 \mathrm{ft}\) of ascent. How much work is performed by the farmer in climbing through a vertical distance of \(60 \mathrm{ft}\) in exactly four revolutions? [Hint: Find a vector field that represents the force exerted by the farmer in lifting his own weight plus the weight of the sack upward at each point along his path.]

Use a CAS to evaluate the line integrals along the given curves.$$ \begin{aligned} &\text { (a) } \int_{C}\left(x^{3}+y^{3}\right) d s \\ &\quad C: \mathbf{r}(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j} \quad(0 \leq t \leq \ln 2) \\ &\text { (b) } \int_{C} x e^{z} d x+(x-z) d y+\left(x^{2}+y^{2}+z^{2}\right) d z \\ &C: x=\sin t, y=\cos t, \quad z=t \quad(0 \leq t \leq \pi / 2) \end{aligned} $$

Let \(C\) be the curve represented by the equations $$ x=t, \quad y=3 t^{2}, \quad z=6 t^{3} \quad(0 \leq t \leq 1) $$ In each part, evaluate the line integral along \(C\). (a) \(\int_{C} x y z^{2} d s\) (b) \(\int_{C} x y z^{2} d x\) (c) \(\int_{C} x y z^{2} d y\) (d) \(\int_{C} x y z^{2} d z\)

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) along the curve \(C\).$$ \begin{aligned} &\mathbf{F}(x, y)=x^{2} y \mathbf{i}+4 \mathbf{j} \\ &C: \mathbf{r}(t)=e^{t} \mathbf{i}+e^{-t} \mathbf{j} \quad(0 \leq t \leq 1) \end{aligned} $$

Use the Divergence Theorem to find the flux of \(\mathbf{F}\) across the surface \(\sigma\) with outward orientation.\(\mathbf{F}(x, y, z)=x^{3} \mathbf{i}+x^{2} y \mathbf{j}+x y \mathbf{k} ; \sigma\) is the surface of the solid bounded by \(z=4-x^{2}, y+z=5, z=0\), and \(y=0\).
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