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17.5 Surface Integrals of Vector Fields
MAT 3724 Applied Analysis I 17.5 Surface Integrals of Vector Fields
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Homework WA
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Preview Surface integral of vector fields
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Recall: Surface Integral of f over the Surface S
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Surface Integrals of Vector Fields
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Physical Considerations…
A fluid is passing through S velocity fun. = v(x,y,z) What is the volume of fluid cross S?
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Physical Considerations…
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Another Example: Heat Flow
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Surface Integral of F over the Surface S (Flux of F across S)
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Evaluations We are going to look at the 3 different type of surfaces and the corresponding formula for the unit normal vector n.
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1. Parametric Surface
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Why is it believable?
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Particular Case 𝑟(𝑥,𝑦) You have seen one particular case from Section 15.4
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(15.4) Normal Vector at (𝑥, 𝑦, 𝑓(𝑥,𝑦))
Tangent Vectors: Normal Vector:
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2. Surface Given by Graphs
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3. Closed Surface Outward Normal (Points away from the solid)
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Example 1 Evaluate the flux of F across S.
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Problem Solving Strategies
Identify the vector field F Classify the surface(s) Identify the orientation from the problem statements (positive, negative, upward, downward, outward, or inward.) Parametrize the surface if it is not given Use the corresponding formula.
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One Last Formula The surface is given by 𝑓(𝑥,𝑦,𝑧)=0
Example: 𝑥 2 + 𝑦 2 + 𝑧 2 =9
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One Last Formula In 15.8, we see that 𝛻𝑓 is perpendicular to the tangent vector 𝑟’( 𝑡 0 ) for every curve 𝐶 passes through the point So 𝛻𝑓 is a normal vector to the surface 𝑓(𝑥,𝑦,𝑧)=0
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One Last Formula A unit normal vector is Surface Integral
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One Last Formula Most useful in problem involving part of a sphere.
Faster than spherical representation Surface Integral
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