Chapter 14: Q. 39 (page 1107)

Use the Fundamental Theorem of Line Integrals, if applicable, to evaluate the integrals in Exercises 37–44. Otherwise, show that the vector field is not conservative.
$F (x, y, z) = (- z, 1, x)$ , with C the circular helix given by $x = \cos t, y = t, z = \sin t, for 0 \leq t \leq 2 π$ .

Short Answer

Expert verified

The given function is not conservative.

Step by step solution

Step 1. Given Information

Use the Fundamental Theorem of Line Integrals, if applicable, to evaluate the integrals in the given exercises. Otherwise, show that the vector field is not conservative.

$F (x, y, z) = (- z, 1, x)$ , with C the circular helix given by $x = \cos t, y = t, z = \sin t, for 0 \leq t \leq 2 π$ .

Step 2. Firstly checking the given field is conservative or not.

$\frac{d F (x, y, z)}{d z} = \frac{d}{d z} (- z) \frac{d F (x, y, z)}{d x} = \frac{d}{d x} x \frac{d F (x, y, z)}{d z} = - 1 \frac{d F (x, y, z)}{d x} = 1$

Since, $\frac{d F (x, y, z)}{d z} \neq \frac{d F (x, y, z)}{d x}$ , so the given function is conservative.

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Use the Fundamental Theorem of Line Integrals, if applicable, to evaluate the integrals in Exercises 37–44. Otherwise, show that the vector field is not conservative.
$F (x, y, z) = (- z, 1, x)$ , with C the circular helix given by $x = \cos t, y = t, z = \sin t, for 0 \leq t \leq 2 π$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Firstly checking the given field is conservative or not.

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Use the Fundamental Theorem of Line Integrals, if applicable, to evaluate the integrals in Exercises 37–44. Otherwise, show that the vector field is not conservative.F(x,y,z)=(−z,1,x), with C the circular helix given by x=cost,y=t,z=sint,for0≤t≤2π.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Firstly checking the given field is conservative or not.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Calculus

Theoretical and Mathematical Physics

Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Use the Fundamental Theorem of Line Integrals, if applicable, to evaluate the integrals in Exercises 37–44. Otherwise, show that the vector field is not conservative.
$F (x, y, z) = (- z, 1, x)$ , with C the circular helix given by $x = \cos t, y = t, z = \sin t, for 0 \leq t \leq 2 π$ .