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Evaluate the surface integral || F·dS for the vector field F(x, y,z)=3yi– 3xj+6zk where S S consists of the hemisphere x +y² +z° = 4 , z> 0 and the disk x² + y° <4, z = 0 with negative orientation. %3D

Evaluate the surface integral || F·dS for the vector field F(x, y,z)=3yi– 3xj+6zk where S S consists of the hemisphere x +y² +z° = 4 , z> 0 and the disk x² + y° <4, z = 0 with negative orientation. %3D

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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**Evaluate the Surface Integral** Given: Evaluate the surface integral \(\iint_{S} \mathbf{F} \cdot d\mathbf{S}\) for the vector field \(\mathbf{F}(x, y, z) = 3y\mathbf{i} - 3x\mathbf{j} + 6z\mathbf{k}\). **Surface \( S \) Description:** - \( S \) consists of two parts: 1. The hemisphere defined by \( x^2 + y^2 + z^2 = 4 \) where \( z \geq 0 \). 2. The disk in the plane \( z = 0 \) defined by \( x^2 + y^2 \leq 4 \). **Orientation:** - The hemisphere has a negative orientation, meaning the normal vectors are directed inward. **Task:** - Calculate the above integral \(\iint_{S} \mathbf{F} \cdot d\mathbf{S}\) using the given conditions and orientation.
Transcribed Image Text:**Evaluate the Surface Integral** Given: Evaluate the surface integral \(\iint_{S} \mathbf{F} \cdot d\mathbf{S}\) for the vector field \(\mathbf{F}(x, y, z) = 3y\mathbf{i} - 3x\mathbf{j} + 6z\mathbf{k}\). **Surface \( S \) Description:** - \( S \) consists of two parts: 1. The hemisphere defined by \( x^2 + y^2 + z^2 = 4 \) where \( z \geq 0 \). 2. The disk in the plane \( z = 0 \) defined by \( x^2 + y^2 \leq 4 \). **Orientation:** - The hemisphere has a negative orientation, meaning the normal vectors are directed inward. **Task:** - Calculate the above integral \(\iint_{S} \mathbf{F} \cdot d\mathbf{S}\) using the given conditions and orientation.
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