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Let $\mathbf{F}$ be a field whose components have continuous first partial derivatives throughout a portion of space containing a region $D$ bounded by a smooth closed surface $S .$ If $|\mathbf{F}| \leq 1,$ can any bound be placed on the size of$$\iiint_{D} \nabla \cdot \mathbf{F} d V ?$$Give reasons for your answer.

The integral's value never exceeds the surface area of $S .$

Calculus 1 / AB

Calculus 3

Chapter 15

Integrals and Vector Fields

Section 8

The Divergence Theorem and a Unified Theory

Integrals

Vectors

Vector Functions

Harvey Mudd College

Baylor University

Idaho State University

Boston College

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

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"].

03:59

Let $\mathbf{F}$ be a diff…

01:54

If $f$ and $g$ are scalar …

07:10

Let $f$ and $g$ have conti…

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