1. 17.5 Surface Integrals of Vector Fields
MAT 3724 Applied Analysis I
17.5
Surface Integrals of Vector Fields

2. Homework WA

3. Preview
Surface integral of vector fields

4. Recall: Surface Integral of f over the Surface S

5. Surface Integrals of Vector Fields

6. Physical Considerations…
A fluid is passing through S
velocity fun. = v(x,y,z)
What is the volume of fluid cross S?

7. Physical Considerations…

8. Another Example: Heat Flow

9. Surface Integral of F over the Surface S (Flux of F across S)

10. Evaluations
We are going to look at the 3 different type of surfaces and the corresponding formula for the unit normal vector n.

11. 1. Parametric Surface

12. Why is it believable?

13. Particular Case 𝑟(𝑥,𝑦)
You have seen one particular case from Section 15.4

14. (15.4) Normal Vector at (𝑥, 𝑦, 𝑓(𝑥,𝑦))
Tangent Vectors:
Normal Vector:

15. 2. Surface Given by Graphs

16. 3. Closed Surface
Outward Normal (Points away from the solid)

17. Example 1
Evaluate the flux of F across S.

18. 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.

19. One Last Formula The surface is given by 𝑓(𝑥,𝑦,𝑧)=0
Example: 𝑥 2 + 𝑦 2 + 𝑧 2 =9

20. 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

21. One Last Formula
A unit normal vector is
Surface Integral

22. One Last Formula Most useful in problem involving part of a sphere.
Faster than spherical representation
Surface Integral