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Show that the curl of $$\mathbf{F}=\frac{-y}{x^{2}+y^{2}} \mathbf{i}+\frac{x}{x^{2}+y^{2}} \mathbf{j}+z \mathbf{k},$$ is zero but that$$\oint_{C} \mathbf{F} \cdot d \mathbf{r}$$.is not zero if $C$ is the circle $x^{2}+y^{2}=1$ in the $x y$ -plane. (Theorem 7 does not apply here because the domain of $\mathbf{F}$ is not simply connected. The field $\mathbf{F}$ is not defined along the $z$ -axis so there is no way to contract $C$ to a point without leaving the domain of $\mathbf{F}$.)

Calculus 3

Chapter 16

Integrals and Vector Fields

Section 7

Stokes’ Theorem

Vector Functions

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

03:04

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x. The input of a function is called the argument and the output is called the value. The set of all permitted inputs is called the domain of the function. Similarly, the set of all permissible outputs is called the codomain. The most common symbols used to represent functions in mathematics are f and g. The set of all possible values of a function is called the image of the function, while the set of all functions from a set "A" to a set "B" is called the set of "B"-valued functions or the function space "B"["A"].

08:32

In mathematics, vector calculus is an important part of differential geometry, together with differential topology and differential geometry. It is also a tool used in many parts of physics. It is a collection of techniques to describe and study the properties of vector fields. It is a broad and deep subject that involves many different mathematical techniques.

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Show that the curl of$…

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Zero curl, yet the field i…

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The vector field $\mathbf{…

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