Welcome to Math 251 – fun in several variables!
In this course, we will generalize the notions of derivatives and integrals from single-variable calculus, but in higher dimensions. Witness the elegance of math, and enjoy the 3D ride!
Syllabus
Math 251 Syllabus
Suggested Problems
Here is a list of Suggested Problems from the textbook, in case would like more practice for the quizzes and exams.
Lecture Notes and Videos
| # |
Section |
Title |
Videos |
| 1 |
12.1 |
3D Coordinate Systems |
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| 2 |
12.2, 12.3 |
Vectors and Dot Products |
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| 3 |
12.4 |
The Cross Product |
Determinants and Bomberman |
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Area of a Parallelogram |
| 4 |
12.5 |
Fun with Lines |
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| 5 |
12.5 |
Fun with Planes |
Distance between Point and Plane |
| 6 |
13.1, 13.2 |
Vector Functions (I) |
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| 7 |
13.2, 13.3 |
Vector Functions (II) |
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| 8 |
13.4, 12.6 |
Vector Functions (III) |
Kamehameha Equation |
| 9 |
14.1 |
Functions of Several Variables |
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| 10 |
14.3 |
Partial Derivatives |
Partial Derivatives |
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Chain Rule Surprise |
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|
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Implicit Differentiation |
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Clairaut Counterexample (optional) |
| 11 |
|
Midterm 1 Review |
|
| 12 |
|
Midterm 1 (Chapters 12, 13) |
|
| 13 |
14.4 |
Tangent Planes |
Tangent Planes |
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Linear Approximations |
| 14 |
14.5 |
The Chain Rule |
|
| 15 |
14.6 |
The Gradient and Applications |
Sum of intercepts |
| 16 |
14.7 |
Max and Min Values (I) |
The true second-derivative test (optional) |
| 17 |
14.7, 14.8 |
Max and Min Values (II)Lagrange Multipliers (I) |
Ikea Problem |
| 18 |
14.8 |
Lagrange Multipliers (II) |
Another Ikea Problem |
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|
|
Arithmetic-Geometric Mean Inequality |
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|
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Distance Point and Plane Lagrange |
| 19 |
15.1, 15.2 |
Double Integrals (I) |
What is a Double Integral? |
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|
Volume of a Ravioli |
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|
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Fubini Counterexample (optional) |
| 20 |
|
Midterm 2 Review |
|
| 21 |
|
Midterm 2 (Chapter 14) |
|
| 22 |
15.1, 15.2 |
Double Integrals (II) |
Changing order of integration |
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|
|
Multivariable Integral |
| 23 |
15.1, 15.2 |
Double Integrals (III) |
|
| 24 |
15.3 |
Double Integrals in Polar Coordinates |
Polar Integral |
|
|
|
Integral sin^2 (x) |
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|
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Volume of an ice cream cone |
|
|
|
Integral over a ring |
| 25 |
15.6 |
Triple Integrals (I) |
Gaussian Integral |
|
|
|
Integral sin(x^2) (optional) |
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|
|
Triple Integrals |
| 26 |
15.6 |
Triple Integrals (II) |
Volume of Gouda Cheese |
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Integral over Cannoli |
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Intersection of 2 cylinders |
| 27 |
15.7 |
Cylindrical Coordinates |
Cylindrical Integral |
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Integral over Princess Cake |
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Integral over SIMS |
| 28 |
15.8 |
Spherical Coordinates (I) |
Spherical Coordinates Derivation |
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Spherical Coordinates Integral |
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Volume of an ice cream cone |
|
|
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Gauss Cubed (optional) |
| 29 |
15.8, 15.9 |
Spherical Coordinates (II)The Jacobian (I) |
Mass of the sun |
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The Jacobian 1 |
| 30 |
15.9 |
The Jacobian (II) |
The Jacobian 2 |
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Change of Variables |
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|
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Polar Coordinate Integral |
|
|
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The Jacobian 3 (optional) |
| 31 |
16.1 |
Vector Fields |
Vector Calculus Overview (optional) |
| Extra |
|
Midterm 3 Review |
|
| 32 |
16.2 |
Line Integrals (I) |
Line Integral 1 |
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Line Integral 2 |
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Line Integral 3 |
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Line Integral 4 |
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Line Integral 5 |
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Line Integral Derivation |
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Line Integral with respect to x |
| 33 |
|
Midterm 3 (Chapter 15) |
|
| 34 |
16.2, 16.3 |
Line Integrals (II)FTC for Line Integrals (I) |
Line Integral of a Vector Field |
| 35 |
16.3 |
FTC for Line Integrals (II) |
FTC Example 1 |
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FTC Example 2 |
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FTC Example 3 |
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Path-independence |
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Don’t use FTC |
| 36 |
16.4 |
Green’s Theorem |
Green’s Theorem |
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Green’s Theorem 2 |
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Area of ellipse |
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|
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Area of polygon |
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|
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Winding Number (optional) |
| 37 |
16.5 |
Curl and Divergence |
|
| 38 |
16.6 |
Parametric Surfaces (I) |
Parametric Surfaces |
|
|
|
Tangent Plane to a Surface |
| 39 |
16.6, 16.7 |
Parametric Surfaces (II)Surface Integrals (I) |
Surface Area of a Sphere |
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|
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Surface area of a donut (optional) |
|
|
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Surface Integral 1 |
|
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Surface Integral 2 |
| 40 |
16.7 |
Surface Integrals (II) |
Surface Integral of a Vector Field |
| 41 |
16.9 |
Divergence Theorem |
Divergence Theorem 1 |
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Divergence Theorem 2 |
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Stokes’ Theorem 1 |
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Integral over a barrel |
| 42 |
16.8 |
Stokes’ Theorem |
Stokes’ Theorem 2 |
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Stokes’ Theorem 3 |
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Crazy Calculus Problem (optional) |
| Extra |
|
Final Exam Review 1 |
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Final Exam Review 2 |
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Quizzes
Here is the schedule of quizzes, which are always due on 11:55 pm (Texas Time). You will take the quizzes on Canvas, except for the first quiz, which will be in class.
Important: There will always be a WebAssign homework due on the same day as the quiz, covering the same sections.
YouTube Playlists and more
Here are the YouTube playlists for this course. Use them in case you would like extra examples
Exams
Here you can find info about exams, as well as other goodies such as practice exams.
Midterm 1: Friday, September 24, covers Chapters 12 and 13
Midterm 2: Friday, October 15, covers Chapter 14
Midterm 3: Friday, November 12, covers Chapter 15
Final Exam: Covers Chapter 16 only
Section 501: Monday, December 13, 10:30 – 12:30 pm
Section 512: Tuesday, December 14, 3:30-5:30 pm