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MIT Calculus III: Surface Integrals, Divergence Theorem, and Stokes' Theorem Worksheet, Study notes of Vector Analysis

Massachusetts Institute of Technology (MIT)Vector Analysis

A worksheet for Interphase Calculus III course at Massachusetts Institute of Technology. The worksheet covers topics such as surface integrals, divergence theorem, and Stokes’ theorem. The questions in the worksheet involve sketching a parameterized surface, calculating the flux of a vector field across a surface, applying the divergence theorem, and verifying the divergence theorem for a given vector field. instructions and formulas for each question.

Typology: Study notes

2021/2022

Uploaded on 05/11/2023

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Interphase Calculus III Worksheet
Instructor: Samuel S. Watson
01 August 2016
Topics: Surface integrals, divergence theorem, and Stokes’ theorem
1. We’ve learned that a parameterized curve in three dimensions is a map rfrom Ito R3, where Iis
an interval in R1. Likewise, parameterized surface is a map rfrom Dto R3where Dis a region in
R2. Sketch the surface r(u,v) = (u,u2,v)where (u,v)ranges over the unit square [0, 1]2.
2. The flux of a vector field Fthrough a surface Sis, if we interpret Fas a velocity field for a fluid,
the amount of fluid passing through the surface per unit time. Writing nfor the unit normal to the
surface at each point, the flux is calculated using the formula
ZZS
F·ndS =ZZD
F·(ru×rv)du dv.
Calculate the flux of F(x,y,z) = (y,xz,x)across the surface from the previous question.
pf3
pf4

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Interphase Calculus III Worksheet Instructor: Samuel S. Watson 01 August 2016

Topics: Surface integrals, divergence theorem, and Stokes’ theorem

  1. We’ve learned that a parameterized curve in three dimensions is a map r from I to R^3 , where I is an interval in R^1. Likewise, parameterized surface is a map r from D to R^3 where D is a region in R^2. Sketch the surface r (u, v) = (u, u^2 , v) where (u, v) ranges over the unit square [0, 1]^2.
  2. The flux of a vector field F through a surface S is, if we interpret F as a velocity field for a fluid, the amount of fluid passing through the surface per unit time. Writing n for the unit normal to the surface at each point, the flux is calculated using the formula ∫∫

S

F · n dS =

∫∫

D

F · ( r u × r v) du dv.

Calculate the flux of F (x, y, z) = (y, xz, x) across the surface from the previous question.

  1. The divergence theorem says that you can calculate the flux of a vector field F out of a closed

surface S two equivalent ways. (1) Using a surface integral:

∫∫

S

F · d S or (2) integrating the diver-

gence over the interior E of the region:

∫∫∫

E

div F dV.

  1. Verify the divergence theorem in the case F (x, y, z) = (x, y, z) over the unit sphere.
  1. Verify the divergence theorem for the vector field (x^2 , xy, z) and the solid bounded by the paraboloid z = 4 − x^2 − y^2 and the xy-plane.
  2. Suppose that G = curl F for some vector field F. Explain why the flux of G through the two surfaces shown below are equal.
  3. (Challenge Problem) Use Stokes’ theorem to find a vector field that can be integrated around the boundary of a region to find the area of the region!