• week 1
  • lecture 1, 8/24/09 vectors in R^n, geometry and algebra, dot product, Cauchy-Schwarz, triangle inequality.
  • recitation 8/25/09 standard basis vectors in R^2, R^3, equations of lines
  • lecture 2, 8/26/09 dot product, Cauchy-Schwarz, triangle inequality, matrices, determinants.
  • recitation, 8/27/09 vector projections
  • lecture 3, 8/28/09 cross product, matrices and determinants (1.4)

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    homework: read 1.1-1.4;  problems: 1.1 #4c, 14a, 16; 1.2 #10, 11, 14, 29; 1.3 #7, 10, 11. 24

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• week 2
  • lecture 4, 8/31/09 Matrices define linear maps and any linear map from R^n to R^m defines a unique matrix. This is very important. The columns of the matrix corresponding to a linear map L are the images under L of the standard basis vectors. Matrix multiplication is defined the way it is so that it corresponds to the composition of linear maps.
    Determinants. What do they measure? They measure the (signed) volume of the image of the unit cube. Various ways of computing the determinants. Properties of the determinants.
  • recitation 9/01/09 cross product, matrices and determinants
  • lecture 5, 9/ 02/09 Limits. It is important to understand what limits are, for otherwise we cannot define differentiability. The official epsilon/delta definition. and its geometric interpretation, in terms of open balls .
  • recitation 9/03/09
  • lecture 6 9/04/09 A function of two variables that had partial derivatives at the origin but was not continuous. The moral is: for a function to be differentiable at a point it's not enough for the partials to exist. The derivative of a function is a linear map that approximates the function ``well."

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    homework: read 1.5, 1.6, 2.2, 2.3; problems (due Wednesday, 9/09/09):
    1.4 # 9, 10, 12, 26; 1.5 #13, 19, 34; 1.6 #2, 7, 11, 21

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• week 3
  • recitation
  • lecture 7 9/09/09 If a function is differentiable then the first order partials exist. Conversely if the first order partials exist and are continuous then the function is differentiable. The geometric meaning of differentiability: the tangent plane to the graph of the function is the graph of the derivative (differential). Finally, two special cases of taking derivatives: (1) the derivative of a real valued function can be thought of as a vector, the gradient vector. (2) the derivative of a curve is the tangent vector.
  • recitation
  • lecture 8 9/11/08 Change of plan: worksheet.

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    homework: read 2.1, 2.2 (lightly), 2.3. problems (due Monday, 9/14/09):
    1.6 #19, 20, 30a; 2.3 # 9, 20, 21, 22, 26, 30, 50.

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• week 4
  • lecture 9 9/14/09 review
  • recitation 9/15/09 FIRST MIDTERM
  • lecture 10 9/16/09 Properties of the derivative, higher order partial derivatives. Chain rule: the derivative (differential) of the composition of two maps is the composition of the differentials.
  • recitation 9/17/09 Chain rule
  • lecture 119/18/09 Directional derivative and gradient

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    homework

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• week 5
  • lecture 12 9/21/09 Inverse and implicit function theorems
  • recitation 9/22/09 Chain rule, implicit function theorem.
  • lecture 13 9/23/09 Inverse function theorem. Vector fields, flow lines and gradient vector fields (2.6 and 3.3, skipping 3.1, 3.2)
  • recitation 9/24/09
  • lecture 14 9/25/09 Divergence, curl, curl (grad f) = 0 and div (curl F) = 0.

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    homework 5 (due Monday, 9/28): read 3.3, 3.4; problems:
    2.6 #20, 38, 44, 45, 46; 3.3 #18, 19.

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• week 6
  • lecture 15 9/28/09 Taylor's theorem (no proofs)
  • recitation 9/29/09 Taylor's theorem
  • lecture 16 9/30/09 More of Taylor's theorem. Maxima and minima, critical points. It is important to understand why maxima and minima are critical points. Hessian as the generalization of the second derivative of the function of one variable. Classification of critical points in terms of Hessians (the test involving principal minors of the corresponding matrix).
  • recitation 10/01/09 Classification of critical points in terms of Hessians
  • lecture 17 10/02/09 Compact sets and the fact that a continuous function on a compact set achieves its maximum and minimum on the set. Constrained maxima and minima and, in particular, the method of Lagrange multipliers for one constraint. It's important to understand why the method works.

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    homework 6 (due Monday, 10/05/09): read 3.4, 3.5, 4.1, 4.2; problems:
    3.3 #25, 3.4 #2, 8, 21, 22, 23, 24, 25, 4.1 #10, 14, 4.2 #1ac, 2ac

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• week 7
  • lecture 18 10/05/09 Lagrange multipliers for 2 constraints. The important points were: the restriction of a function f to a constraint S has an extremal point at P in S if and only if the gradient of f at P is perpendicular to S. If the constraint S is cut out by the functions g_1 and g_2 then at any point Q the set of vectors perpendicular to S are in the span of grad g_1 (Q) and grad g_2 (Q). Therefore to find the extremal points of f on S look for points where grad f is a linear combination of grad g_1 and grad g_2.
  • recitation 10/06/09 Lagrange multipliers
  • lecture 19 10/07/09 Double integrals
  • recitation 10/08/09 Double integrals
  • lecture 20 10/09/09 Double integrals in polar coordinates

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    homework 7 (due Monday, 10/12/09): read 4.2, 4.3, 5.1-5.3 ; problems:
    4.2 #6, 22, 32, 4.3 #6, 7, 8, 25, 5.1 #10, 11, 5.2 #1, 2, 27, 5.3 #2,3.

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• week 8
  • lecture 21 10/12/09 Triple integrals
  • recitation 10/13/09 triple integrals
  • lecture 22 10/14/09 Triple integrals in cylindrical and spherical coordinates
  • recitation 10/15/09 Triple integrals in cylindrical and spherical coordinates
  • lecture 23 10/16/09 Change of variables in double and triple integrals.

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    homework 8 (due Monday, 10/19/21): read 5.3, 5.4; problems:
    5.2 #29, 5.3 #16, 17, 18, 5.4 # 14, 15, 16, 5.5 # 1, 2, 3, 15, 21, 23, 31.

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• week 9
  • lecture 24 10/1/09 review
  • recitation 10/20/09 SECOND MIDTERM
  • lecture 25 10/21/09 Change of variables in double and triple integrals.
  • recitation 10/22/09 Change of variables in double and triple integrals.
  • lecture 26 10/23/09 Line integrals of functions, vector fields and 1-forms. Line integrals of functions are independent of the parameterization. Line integrals of of vector fields depend only on the direction in which the paths are traversed and not on the parameterization itself. Reversing the direction of the path changes the sign of the integral.

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    homework (due Monday 10/26/09): read 5.5, 6.1; problems:
    5.5 #7, 8, 9, 10, 11; 6.1 #2, 6, 15, 16, 17

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• week 10
  • lecture 27 10/26/09 Covectors, 1-forms as covector fields. Geometry of integration of 1-forms over curves: you feed a tangent vector to the curve into a 1-form and get a function. You then integrate the function. This is not really in the book, but check chapter 8 for more on differential forms. Correspondence between 1-forms and vector fields. Under the correspondence gradients correspond to total differentials.
  • recitation 10/27/09 line integrals
  • lecture 28 10/28/09 Green's theorem for vector fields and for differential forms. 2-forms and exterior derivatives of 1-forms.
  • recitation 10/29/09 Green's theorem
  • lecture 29 10/30/09 Proof of Green's theorem. Green's theorem in terms of curl. The divergence theorem for vector fields in the plane.

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    homework(due Monday 11/02/09): read 6.1, 6.2, 6.3, 8.1; problems:
    6.1 #20, 22, 24, 25a, 27, 29; 6.2 # 1, 2, 7, 8, 10, 20

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• week 11
  • lecture 30 11/02/09 Two theorems on the dependence of line integrals on the path. Simply-connected regions.
  • recitation 11/03/09 Green's theorem.
  • lecture 31 11/04/09 Proofs of theorem 3.3 and theorem 3.5 of section 6.3 in textbook.
  • recitation 11/05/09 Finding a potential of a conservative vector field.
  • lecture 32 11/06/09 Parameterizations of surfaces. Surface integrals of functions.

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    homework: read 6.3, 8.1, 7.1; problems:
    6.2 # 21, 22, 23, 24, 25; 6.3 #1, 3, 4, 5, 10, 15, 17.

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• week 12
  • lecture 33 11/09/09 Normal vectors to surfaces, orientability of surfaces. Flux integral.
  • recitation 11/10/09 Surface integrals of functions and vector fields.
  • lecture 34 11/11/09 From flux integrals to integrals of 2-forms.
  • recitation 11/12/09 Surface integrals.
  • lecture 35 11/13/09 Surface integrals of 2-forms.

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    homework: read 7.1, 7.2; problems:
    7.1 #1a, 4ac, 8, 12; 7.2 #1, 2, 3, 4, 10, 12.

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    solutions are here

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• week 13
  • lecture 36 11/16/09 review
  • recitation 11/17/09 THIRD MIDTERM (take home)
  • lecture 37 11/18/09 Stokes and Gauss (divergence) theorems.
  • recitation 11/19/09
  • lecture 38 11/20/09 Differential forms and the dictionary. my notes of the lecture

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    suggested homework: read 7.3, 8.1; problems:
    7.2 #20, 22; 7.3 #1, 3, 7, 9, 12, 14

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• Thanksgiving break week
  • lecture Thanksgiving break
  • recitation Thanksgiving break
  • lecture Thanksgiving break
  • recitation Thanksgiving break
  • lecture Thanksgiving break

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• week 14
  • lecture 39 11/30/09 Pullback of differential forms.
  • recitation 12/01/09 Suggested homework problems and pullback problems.
  • lecture 40 12/02/09 orientation of regions of R^n my notes of the lecture
  • recitation 12/03/09 Here are some notes
  • lecture 41 12/04/09 integration of k-forms over k-manifolds and generalized Stokes' theorem my notes of the lecture

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    last homework: read 8.1, 8.2, 8.3; problems:
    8.1 #6,7, 10, 11, 15; 8.2 #6, 7, 9, 12, 13; 8.3 #4, 5, 6, 10, 11, 12.

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• week 15
  • lecture 42 12/07/09 Generalized Stokes' theorem
  • recitation review
  • lecture 12/09/09 review (2nd version)

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    homework: read ;

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Last modified: Mon Dec 7 16:23:02 CST 2009