MA 261: Multivariate Calculus
Instructor: Katy Yochman
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MA 261 course website for the current semester
Notes (from Summer 2019)
Exam 1 Material
- Lesson 3
§13.6: Cylinders and Quadratic Surfaces
§14.1: Vector-Valued Functions
Lesson 3 Worksheet: Space Curves (with answers)
- Lesson 4
§14.2: Calculus of Vector-Valued Functions
§14.3: Motion in Space
- Lesson 5
§14.3: Motion in Space
§14.4: Length of Curves
§14.5: Curvature and Normal Vectors
- Lesson 6
§15.1: Graphs and Level Curves
§15.2: Limits and Continuity
- Lesson 7
§15.3: Partial Derivatives
§15.4: The Chain Rule
- Lesson 8
§15.5: Directional Derivatives and the Gradient
§15.6: Tangent Planes and Linear Approximations
See figures on Pg 965 - 968 for the gradient
See figures on Pg 974 - 976 for the tangent plane to a surface.
Exam 2 Material
- Lesson 9
§15.7: Maximum/Minimum Problems
Note: Example 4 is incomplete. When we check the boundary in Step 2, we need to check the ENDPOINTS as well. For instance, when we set y equal to 2, we get a function of x, f(x,2) = x^4 + 16 - 8x. Since we are looking for an ABSOLUTE min or max, we need to recall the Extreme Value Theorem from Calc 1 where we take the derivative and check critical points in the interval as well as the endpoints of the interval. In the notes, we found the critical point (2^(1/3), 2), on the part of the boundary y = 2, but we ALSO need to check the endpoints of the interval (-2, 2) and (2,2).
- Lesson 10
§15.8: Lagrange Multipliers
- Lesson 11
§16.1: Double Integrals over Rectangular Regions
- Lesson 12
§16.2: Double Integrals over General Regions
- Lesson 13
§16.3: Double Integrals in Polar Coordinates
The last page includes a list of trig identities you should know.
- Lesson 14
§16.4: Triple Integrals
- Lesson 15
§16.5 Triple Integrals in Cylindrical and Spherical Coordinates
- Lesson 16 (Part 1)
§16.6: Integrals for Mass Calculations
- Lesson 16 (Part 2)
§17.1: Vector Fields
- Lesson 17
§17.2: Line Integrals (or Integrals over a Curve)
Note: Flux is NOT just over closed curves. Flux can be taken along ANY curve.
Final Exam Material
- Lesson 18
§17.3: Conservative Vector Fields
- Lesson 19 (Part 1)
§17.4: Green's Theorem
- Lesson 19 (Part 2)
§17.5: Divergence and Curl
- Lesson 20
§17.6: Surface Integrals - surface parameterization and integrals of functions of several variables
To emphasize the difference between a parameterization (which we are using) and a projection of the sphere (which is what a map is), here's a helpful video: Why all world maps are wrong
- Lesson 21
§17.6: Surface Integrals - explicitly defined surfaces
- Lesson 22
§17.6: Surface Integrals - oriented surfaces and integrals of vector fields
- Lesson 23
§17.7: Stokes' Theorem
- Lesson 24
§17.7: Stokes' Theorem
- Lesson 25
§17.8: Divergence Theorem
- Lesson 26
§17.8: Divergence Theorem
Review