Chapter 12: Q. 17. (page 964)

Sketch level curves $z = - 1, 0$ , and 1 for the function
$z = x^{2} - y^{2}$ . Include the graphs of three gradient vectors
on each level curve. What do you observe?

Short Answer

Expert verified

The graph of level curves $z = x^{2} - y^{2}$ for is shown below:

The gradient and tangent vectors are orthogonal.

Step by step solution

The objective is to sketch the level curves z=-1,0,1 for the function

The function $z = x^{2} - y^{2} .$

The graph of the function $z = x^{2} - y^{2}$ is a concentric circle, each of whose level curves is a circle centered at the origin.

The gradient is,
$z = x^{2} - y^{2} \nabla z = 2 x i - 2 y j$

Hence, the gradient vectors are perpendicular to the level curves and point toward the $x$ -axis while avoiding the $y$ -axis. The magnitude of the gradient vectors grows.

Draw a graph of level curves z=x2-y2 for z=-1,0,1 and write the observation

The level of curves $z = x^{2} - y^{2}$ for $z = - 1, 0, 1$ is

$- 1 = x^{2} - y^{2} 0 = x^{2} - y^{2} 1 = x^{2} - y^{2}$

The graph of level curves $z = x^{2} - y^{2}$ for $z = - 1, 0, 1$ is shown below:

Therefore, the gradient and tangent vectors are orthogonal.

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Sketch level curves $z = - 1, 0$ , and 1 for the function
$z = x^{2} - y^{2}$ . Include the graphs of three gradient vectors
on each level curve. What do you observe?

Short Answer

Step by step solution

The objective is to sketch the level curves z=-1,0,1 for the function

Draw a graph of level curves z=x2-y2 for z=-1,0,1 and write the observation

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Sketch level curvesz=-1,0, and 1 for the functionz=x2-y2. Include the graphs of three gradient vectorson each level curve. What do you observe?

Short Answer

Step by step solution

The objective is to sketch the level curves z=-1,0,1 for the function

Draw a graph of level curves z=x2-y2 for z=-1,0,1 and write the observation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Mechanics Maths

Decision Maths

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Sketch level curves $z = - 1, 0$ , and 1 for the function
$z = x^{2} - y^{2}$ . Include the graphs of three gradient vectors
on each level curve. What do you observe?