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Chapter 12: Q. 16 (page 987)

Give precise mathematical definitions or descriptions of each of the following concepts that follow. Then illustrate the definition with a graph or an algebraic example.

The gradient of a function fof two or three variables.

Short Answer

Expert verified

Let z=f(x,y)be a function of two variables. The gradient of fis the vector function defined by鈭�f(x,y)=鈭�f鈭�xi+鈭�f鈭�yj=fx(x,y),fy(x,y)

Similarly, if w=f(x,y,z)is a function of three variables, then the gradient of fis the vector function defined by 鈭�f(x,y,z)=鈭�f鈭�xi+鈭�f鈭�yj+鈭�f鈭�zk=fx(x,y,z),fy(x,y,z),fz(x,y,z)

Step by step solution

01

Step 1. Given information   

The gradient of a function fof two or three variables.

02

Step 2. Defining the gradient of a function f

Let z=f(x,y)be a function of two variables. The gradient of fis the vector function defined by鈭�f(x,y)=鈭�f鈭�xi+鈭�f鈭�yj=fx(x,y),fy(x,y)

Similarly, if w=f(x,y,z)is a function of three variables, then the gradient of fis the vector function defined by 鈭�f(x,y,z)=鈭�f鈭�xi+鈭�f鈭�yj+鈭�f鈭�zk=fx(x,y,z),fy(x,y,z),fz(x,y,z)

The domain of the gradient is the set of all points in the domain of fat which the partial derivatives exist.

Example:

The gradient of the function,f(x,y)=x2yis:鈭�f(x,y)=鈭�f鈭�xi+鈭�f鈭�xj=2xyi鈭�x2y2j

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Most popular questions from this chapter

In Example 4 we found that the function f(x,y)=x3+9x2+6y2has stationary points at (0,0)and (-6,0).

(a) Use the second-derivative test to show that \(f\) has a saddle point at(-6,0).

(b) Use the second-derivative test to show that \(f\) has a relative minimum at(0,0).

(c) Use the value of \(f(-10,0)\) to argue that \(f\) has a relative minimum at (0,0),and not an absolute minimum, without using the second-derivative test.

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.

f(x,y)=xyx2+y2,P=(鈭�2,1),v=35,鈭�45

Use Theorem 12.32 to find the indicated derivatives in Exercises 21鈥�26. Express your answers as functions of a single variable.

dydtwheny=rsin胃,r=t3,and胃=t

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:

f(x,y,z)=xz2ey

Describe the meanings of each of the following mathematical expressions:

鈭�f(x,y)
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