Chapter 14: Q. 23 (page 1131)

$Compute the curl of the vector fields in Exercises 23 - 28 . F (x, y, z) = \sin^{- 1} (xy) i + \ln (y + z) j + \frac{1}{2 x + 3 y + 5 z + 1} k$

Short Answer

Expert verified

$curl F = - (\frac{3}{{(2 x + 3 y + 5 z + 1)}^{2}} + \frac{1}{y + z}) i + \frac{2}{{(2 x + 3 y + 5 z + 1)}^{2}} j - \frac{x}{\sqrt{1 - {(xy)}^{2}}} k$

Step by step solution

Step 1. Given information is:

$F (x, y, z) = \sin^{- 1} (xy) i + \ln (y + z) j + \frac{1}{2 x + 3 y + 5 z + 1} k$

Step 2. Curl of vector field

$Here, F (x, y, z) = \sin^{- 1} (xy) i + \ln (y + z) j + \frac{1}{2 x + 3 y + 5 z + 1} k Compare this with, F (x, y, z) = f_{1} (x, y, z) i + f_{2} (x, y, z) j + f_{3} (x, y, z) k Then, f_{1} (x, y, z) = \sin^{- 1} (xy), f_{2} (x, y, z) = \ln (y + z), f_{3} (x, y, z) = \frac{1}{2 x + 3 y + 5 z + 1} Now, curl F = |\begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \sin^{- 1} (xy) & \ln (y + z) & \frac{1}{2 x + 3 y + 5 z + 1} \end{matrix}| = i (\frac{\partial (\frac{1}{2 x + 3 y + 5 z + 1})}{\partial y} - \frac{\partial (\ln (y + z))}{\partial z}) + j (\frac{\partial (\sin^{- 1} (xy))}{\partial z} - \frac{\partial (\frac{1}{2 x + 3 y + 5 z + 1})}{\partial x}) + k (\frac{\partial (\ln (y + z))}{\partial x} - \frac{\partial (\sin^{- 1} (xy))}{\partial y}) = - (\frac{3}{{(2 x + 3 y + 5 z + 1)}^{2}} + \frac{1}{y + z}) i + \frac{2}{{(2 x + 3 y + 5 z + 1)}^{2}} j - \frac{x}{\sqrt{1 - {(xy)}^{2}}} k$

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$Compute the curl of the vector fields in Exercises 23 - 28 . F (x, y, z) = \sin^{- 1} (xy) i + \ln (y + z) j + \frac{1}{2 x + 3 y + 5 z + 1} k$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Curl of vector field

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ComputethecurlofthevectorfieldsinExercises23–28.F(x,y,z)=sin-1(xy)i+ln(y+z)j+12x+3y+5z+1k

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Curl of vector field

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

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Study anywhere. Anytime. Across all devices.

$Compute the curl of the vector fields in Exercises 23 - 28 . F (x, y, z) = \sin^{- 1} (xy) i + \ln (y + z) j + \frac{1}{2 x + 3 y + 5 z + 1} k$