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Find the divergence of the field.$$\mathbf{F}=y e^{x y z} \mathbf{i}+z e^{x y z} \mathbf{j}+x e^{x z} \mathbf{k}$$

$$\left(y^{2} z+x z^{2}+x^{2} y\right) e^{x y z}$$

Calculus 1 / AB

Calculus 3

Chapter 15

Integrals and Vector Fields

Section 8

The Divergence Theorem and a Unified Theory

Integrals

Vectors

Vector Functions

Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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"].

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Find the divergence of the…

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our goal is to find the emergence of which we know to be equal to the partial derivative with respect to acts are first term, which in this case, is why the X y plus the partial through our second term with respect to y which is Z e to the X y z no all right below the third term so we can fit it all plus de over the easy this time times x e x qy So now we saw for our partial derivatives So our first term are wise gonna be a constant with respect to X And then we have an exponential function with the axe. So that's just itself Times the train chain room, which is just y z You do this pretty much the same thing for the next few. RC is gonna be a constant with respect to why exponential. So we get the same exponents, then times the chain will which in this case is gonna be X. All right, the last term below it. So little. So our axes a constant our export. It stays the same that our chain role with respect to see just gonna be X and then not to read it. In our simplest terms, we can factor out and e to the X y Z and that we just battle over terms together. So we have Why square times he from our first term plus Z squared times tax for a second term, plus X squared times. Why from our third term and we're left with this answer.

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