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Assuming that all the necessary derivatives exist and are continuous, show that if$f(x, y)$ satisfies the Laplace equation $$\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0.$$ then $$\oint_{C} \frac{\partial f}{\partial y} d x-\frac{\partial f}{\partial x} d y=0$$ for all closed curves $C$ to which Green's Theorem applies. (The converse is also true: If the line integral is always zero, then $f$ satisfies the Laplace equation.)

$$=\int_{\Omega} \int\left(-\frac{\partial^{2} f}{\partial x^{2}}-\frac{\partial^{2} f}{\partial y^{2}}\right) d x d y=0$$

Calculus 3

Chapter 16

Integrals and Vector Fields

Section 4

Green’s Theorem in the Plane

Vector Functions

Campbell University

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University of Michigan - Ann Arbor

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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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Green's Theorem and L…

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Kate. We can't just use that role. We'll just use the green serum to decorate the well. So partial, impartial acts. This is so partial end part of acts which is minus I shall ask partial X right minus partial, partial and partial. Why so minus? This is an ah, partial add partial. Why always should be zero if we are because it satisfies the lettuce. Your question so they grow should be zero. And that is true for all coast curves.

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