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क्ने Find the curl of F(x, y, 2) = (2x,3y, 52). < 2, 3, 5 > depends on the point (x, y, z) ● 10 O < 0, 0, 0 >

क्ने Find the curl of F(x, y, 2) = (2x,3y, 52). < 2, 3, 5 > depends on the point (x, y, z) ● 10 O < 0, 0, 0 >

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Calculus Problem: Computing Curl of a Vector Field #### Problem Statement: Find the curl of \(\vec{F}(x, y, z) = \langle 2x, 3y, 5z \rangle\). #### Answer Choices: - ⭕ `< 2, 3, 5 >` - ⭕ `10` - ⭕ `depends on the point (x, y, z)` - ⭘ `< 0, 0, 0 >` #### Solution Explanation: In vector calculus, the curl of a vector field \(\vec{F} = \langle P, Q, R \rangle\) is given by: \[ \nabla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \] For \(\vec{F}(x, y, z) = \langle 2x, 3y, 5z \rangle\): - \(P = 2x\) - \(Q = 3y\) - \(R = 5z\) Calculate the components of the curl: - \(\frac{\partial R}{\partial y} = \frac{\partial (5z)}{\partial y} = 0\) - \(\frac{\partial Q}{\partial z} = \frac{\partial (3y)}{\partial z} = 0\) - \(\frac{\partial P}{\partial z} = \frac{\partial (2x)}{\partial z} = 0\) - \(\frac{\partial R}{\partial x} = \frac{\partial (5z)}{\partial x} = 0\) - \(\frac{\partial Q}{\partial x} = \frac{\partial (3y)}{\partial x} = 0\) - \(\frac{\partial P}{\partial y} = \frac{\partial (2x)}{\partial y} = 0\) Thus: \[ \nabla \times \vec{F} = \langle 0 - 0, 0 -
Transcribed Image Text:### Calculus Problem: Computing Curl of a Vector Field #### Problem Statement: Find the curl of \(\vec{F}(x, y, z) = \langle 2x, 3y, 5z \rangle\). #### Answer Choices: - ⭕ `< 2, 3, 5 >` - ⭕ `10` - ⭕ `depends on the point (x, y, z)` - ⭘ `< 0, 0, 0 >` #### Solution Explanation: In vector calculus, the curl of a vector field \(\vec{F} = \langle P, Q, R \rangle\) is given by: \[ \nabla \times \vec{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \] For \(\vec{F}(x, y, z) = \langle 2x, 3y, 5z \rangle\): - \(P = 2x\) - \(Q = 3y\) - \(R = 5z\) Calculate the components of the curl: - \(\frac{\partial R}{\partial y} = \frac{\partial (5z)}{\partial y} = 0\) - \(\frac{\partial Q}{\partial z} = \frac{\partial (3y)}{\partial z} = 0\) - \(\frac{\partial P}{\partial z} = \frac{\partial (2x)}{\partial z} = 0\) - \(\frac{\partial R}{\partial x} = \frac{\partial (5z)}{\partial x} = 0\) - \(\frac{\partial Q}{\partial x} = \frac{\partial (3y)}{\partial x} = 0\) - \(\frac{\partial P}{\partial y} = \frac{\partial (2x)}{\partial y} = 0\) Thus: \[ \nabla \times \vec{F} = \langle 0 - 0, 0 -
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