Chapter 14: Q. 24 (page 1132)

$C o m p u t e t h e c u r l o f t h e v e c t o r f i e l d s i n E x e r c i s e s 23 - 28 . G (x, y, z) = (x^{2} + y - z) i - 2 y \cos zj + e^{x^{2} + y^{2} + z^{2}} k$

Short Answer

Expert verified

$curl G = = (2 {ye}^{x^{2} + y^{2} + z^{2}} - 2 y sinz) i + 2 {xe}^{x^{2} + y^{2} + z^{2}} j - k$

Step by step solution

Step 1. Given information is:

$G (x, y, z) = (x^{2} + y - z) i - 2 y \cos zj + e^{x^{2} + y^{2} + z^{2}} k$

Step 2. Curl of vector field

$Here, G (x, y, z) = (x^{2} + y - z) i - 2 y \cos zj + e^{x^{2} + y^{2} + z^{2}} k Compare this with, G (x, y, z) = g_{1} (x, y, z) i + g_{2} (x, y, z) j + g_{3} (x, y, z) k Then, g_{1} (x, y, z) = x^{2} + y - z, g_{2} (x, y, z) = - 2 y \cos z, g_{3} (x, y, z) = e^{x^{2} + y^{2} + z^{2}} Now, curl G = |\begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ x^{2} + y - z & - 2 y \cos z & e^{x^{2} + y^{2} + z^{2}} \end{matrix}| = i (\frac{\partial (e^{x^{2} + y^{2} + z^{2}})}{\partial y} - \frac{\partial (- 2 y \cos z)}{\partial z}) + j (\frac{\partial (x^{2} + y - z)}{\partial z} - \frac{\partial (e^{x^{2} + y^{2} + z^{2}})}{\partial x}) + k (\frac{\partial (- 2 y \cos z)}{\partial x} - \frac{\partial (x^{2} + y - z)}{\partial y}) = (2 {ye}^{x^{2} + y^{2} + z^{2}} - 2 y sinz) i + 2 {xe}^{x^{2} + y^{2} + z^{2}} j - k$

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$C o m p u t e t h e c u r l o f t h e v e c t o r f i e l d s i n E x e r c i s e s 23 - 28 . G (x, y, z) = (x^{2} + y - z) i - 2 y \cos zj + e^{x^{2} + y^{2} + z^{2}} k$

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Curl of vector field

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ComputethecurlofthevectorfieldsinExercises23–28.G(x,y,z)=(x2+y−z)i−2ycoszj+ex2+y2+z2k

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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$C o m p u t e t h e c u r l o f t h e v e c t o r f i e l d s i n E x e r c i s e s 23 - 28 . G (x, y, z) = (x^{2} + y - z) i - 2 y \cos zj + e^{x^{2} + y^{2} + z^{2}} k$