Chapter 14: Q. 16 (page 1095)

How would you show that a given vector field in $ℝ^{3}$ is not conservative?

Short Answer

Expert verified

A given vector field in $ℝ^{3}$ is not conservative when,

$F (x, y, z) \neq \nabla f (x, y, z) \neq \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j + \frac{\partial f}{\partial z} k$

Step by step solution

Step 1.Given Information

How would you show that a given vector field in $ℝ^{3}$ is not conservative?

Step 2. A vector field is conservative

If a vector field can be described as a gradient, it offers a number of mathematically appealing properties, including the ability to be easily integrated along curves. Such fields are referred to as conservative vector fields.

A vector field that is conservative. F is a vector field that represents the gradient of a function $f$ .

Step 3. A conservative vector field F is a vector field that can be written as the gradient of some function f.

That is,

$F (x, y, z) = \nabla f (x, y, z) = \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j + \frac{\partial f}{\partial z} k$

Hence, we can say that a given vector field in $ℝ^{3}$ is not conservative, when

$F (x, y, z) \neq \nabla f (x, y, z) \neq \frac{\partial f}{\partial x} i + \frac{\partial f}{\partial y} j + \frac{\partial f}{\partial z} k$

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How would you show that a given vector field in $ℝ^{3}$ is not conservative?

Short Answer

Step by step solution

Step 1.Given Information

Step 2. A vector field is conservative

Step 3. A conservative vector field F is a vector field that can be written as the gradient of some function f.

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How would you show that a given vector field in ℝ3 is not conservative?

Short Answer

Step by step solution

Step 1.Given Information

Step 2. A vector field is conservative

Step 3. A conservative vector field F is a vector field that can be written as the gradient of some function f.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Probability and Statistics

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

How would you show that a given vector field in $ℝ^{3}$ is not conservative?