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Use a CAS and Green's Theorem to find the counterclockwise circulation of the field $\mathbf{F}$ around the simple closed curve $C$. Perform the following CAS steps. a. Plot $C$ in the $x y$ -plane.b. Determine the integrand $(\partial N / \partial x)-(\partial M / \partial y)$ for the tangential form of Green's Theorem.c. Determine the (double integral) limits of integration from your plot in part (a) and evaluate the curl integral for the circulation.$\mathbf{F}=x^{-1} e^{y} \mathbf{i}+\left(e^{y} \ln x+2 x\right) \mathbf{j}$C: The boundary of the region defined by $y=1+x^{4}$ (below) and $y=2(\text { above })$

Calculus 3

Chapter 16

Integrals and Vector Fields

Section 4

Green’s Theorem in the Plane

Vector Functions

Johns Hopkins University

Harvey Mudd College

Idaho State University

Boston College

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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Use a CAS and Green's…

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