Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Continue

Input your name and email to request the answer

Like

Report

Green's first formula Suppose that $f$ and $g$ are scalar functions with continuous first- and second-order partial derivatives throughout a region $D$ that is bounded by a closed piecewise smooth surface S. Show that $$\iint_{S} f \nabla g \cdot \mathbf{n} d \sigma=\iiint_{D}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V$$ Equation (10) is Green's first formula. (Hint: Apply the Divergence Theorem to the field $\mathbf{F}=f \nabla g .$ )

Calculus 3

Chapter 16

Integrals and Vector Fields

Section 8

The Divergence Theorem and a Unified Theory

Vector Functions

Campbell University

Harvey Mudd College

University of Nottingham

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.

02:22

Green's first formula…

06:07

06:00

01:59

Green's First Identit…

No transcript available

View More Answers From This Book

Find Another Textbook

02:16

Find the slope, $x$ -intercept and $y$ -intercept of the given line.$$4 …

03:47

Two cars leave a city traveling in the same direction. One travels at $60 \m…

03:59

In each of the following exercises, solve the given inequality.$$(2 x+3)…

04:02

In each of the following exercises, solve the given inequality.$$x^{3}-1…

Boston and New York are approximately 280 miles apart. One train leaves Bost…

01:45

Find the unknown.$$(2 y-3)^{2}=49$$

05:03

Today, Maria's age is 5 years less than twice Louie's age. In five…

03:15

Solve each of the quadratics by first completing the square. When the roots …

01:03

Complete the square in each of the following by putting in the form $(x+B)^{…

01:22