Midterm
- Our midterm was the during class on March 15, the Friday before spring break.
- Materials for preparing for the exam: review sheet, solutions to practice problems from the review sheet.
Week 1
- Monday. Course summary.
- Tuesday. Definition of the integral.
- Wednesday. Refinement of partitions.
- Friday. Properties of infs and sups.
Wednesday Quiz:
No quiz this week.Homework due Friday:
- No HW this week.
Week 2
- Monday. Integrability criterion. Uniform continuity.
- Tuesday. Continuous functions are integrable. Fundamental theorem of calculus in one variable. Here is a link to the proof of the mean value theoreom, and to a proof of the fundamental theorem of calculus in one variable.
- Wednesday. Properties of integrals. Integrals over non-boxes. (Here is a proof of a certain property of infs and sups we used today: infs&sups.
- Friday. Continuous except on a set of volume zero \(\Rightarrow\) integrable.
Wednesday Quiz:
Let \(f\colon B\to\mathbb{R}\) be a bounded function on a box \(B\).- What does it mean to say that \(f\) is integrable? (Define the \(m_J(f)\), the \(M_J(f)\), the lower and upper sums, and the lower and upper integrals as part of your definition.)
- What is the key idea behind the proof that \(L(f,P)\leq U(f,Q)\) for any partitions \(P\) and \(Q\) of \(B\)?
Homework due Friday:
- 6.1.1
- 6.1.3 (no proofs required)
- 6.1.4
- 6.2.3
- 6.3.5 (proof required)
- 1. Let \(X\) and \(Y\) be sets of real numbers. Define \[X+Y:=\{x+y:x\in X\text{ and }y\in Y\}.\] Show that if \(\tilde{x}=\sup X\) and \(\tilde{y}=\sup Y\), then \(\tilde{x}+\tilde{y}=\sup (X+Y)\).
- 2. Let \(X\) be a set of real numbers, and let \(c>0\). Define \(cX=\{cx:x\in X\}\). Prove that if \(\tilde{x}=\sup X\), then \(c\tilde{x}=\sup(cX)\).
Week 3
- Monday. Consequence’s of the theorem from our last class. Examples using Fubini’s theorem.
- Tuesday. Review HW.
- Wednesday. Fubini’s theorem, examples.
- Friday. Change of variables theorem.
Wednesday Quiz:
- Let \(B\subset\mathbb{R}^n\) be a box, and let \(f\colon B\to\mathbb{R}\) be a continuous function. Prove that \(f\) is integrable.
- Let \(S\subset\mathbb{R}^n\) be a bounded set, and let \(f\colon S\to\mathbb{R}\) be a bounded function. What does it mean to say that \(f\) is integrable over \(S\)?
Homework due Friday:
- 6.2.7 (Use problem [A], below, and disregard the comment in the text about letting \(f=0\).)
- 6.3.4 (hint: MVT)
- 6.4.3 (hint: consider \(F_2-F_1\), and see the previous hint)
- 6.5.6 (note: it is not obvious that \(\chi_R\) is integrable)
- [A] Let \(X\) be a non-empty subset of \(\mathbb{R}^n\), and let \(f\) and \(g\) be real-valued functions on \(X\). Suppose that \(f(x)\leq g(x)\) for all \(x\in X\) and that \(f\), hence \(g\), is bounded below on \(X\). Prove that \(\inf f(X)\leq \inf g(X)\).
- [B] Evaluate the integrals of the following functions using Fubini’s theorem.
- \(f(x,y) = x^2+y\) on \([0,1]\times[0,2]\)
- \(g(x,y) = x^2+y\) on \(\{(x,y): x\geq0,\ y\geq0,\ x+y\leq1\}\)
- \(h(x,y,z) = x\) on \(\{(x,y,z): x\geq0,\ y\geq0,\ z\geq0,\ x+y+z\leq1\}\).
Week 4
- Monday Determinants.
- Tuesday. Review HW. Explanation of change of coordinates formula. Common changes of coordinates.
- Wednesday. Common changes of coordinates.
- Friday. Volume of the \(n\)-sphere. Practice problems.
Wednesday Quiz:
- Let \(S\subset\mathbb{R}^n\) be a bounded set (not necessarily a box), and let \(f\colon S\to\mathbb{R}\) be a bounded function. What does it mean to say that \(f\) is integrable on \(S\), and in that case, what is \(\int_Sf\)?
- Let \(S\subset\mathbb{R}^n\), and let \(f\colon S\to\mathbb{R}^m\). What does it mean to say that \(f\) is uniformly continuous? How does the notion of uniform continuity differ from that of ordinary continuity? Give an example of a function that is continuous but not uniformly continuous.
Homework due Friday:
- 6.2.8
- 6.6.1
- 6.6.2
- 6.6.3
- 6.6.5
- 6.7.13
Week 5
- Monday. Practice problems.
- Tuesday. Review HW.
- Wednesday. Ordinary tensors.
- Friday. Alternating tensors, differential forms.
Wednesday Quiz:
- State the change of variables formula with its hypotheses.
- State the spherical change of coordinates and draw a picture to show where it comes from. What is the stretching factor for spherical change of coordinates?
- Same problem as above for the cylindrical change of coordinates.
- Be able to use Fubini and changes of coordinates to evaluate integrals.
Homework due Friday:
- 6.7.1
- 6.7.3
- 6.7.8
- 6.7.10
- 6.7.11
- 6.7.14 (abc)
- Let \(v=(1,0,3)\) and \(w=(2,3,-4)\). Write \(v\otimes w\) as a linear combination of tensor products of the standard basis vectors.
Week 6
- Monday. Differential forms; the algebra of forms; the exterior derivative.
- Tuesday. Go over HW. Differential forms. Integration of \(n\)-forms in \(\mathbb{R}^n\); \(k\)-surfaces in \(\mathbb{R}^n\); pullbacks. Not a good day to miss class.
- Wednesday. Integration of \(k\)-forms in \(\mathbb{R}^n\) over \(k\)-surfaces.
- Friday. Basic properties of pullbacks and the exterior derivative.
Wednesday Quiz:
- No quiz this week.
Homework due Friday:
- HW 5.
Week 7
- Monday. Chains.
- Tuesday. Go over HW. Boundaries.
- Wednesday. The boundary of a donut.
- Friday. Midterm: review sheet, solutions to practice problems from the review sheet.
Wednesday Quiz:
- No quiz this week.
Homework due Friday:
- No homework this week.
Week 8
- Monday. Line integrals.
- Tuesday. Return midterms. The flow of a vector field along a curve.
- Wednesday. Gradient vector fields, potentials.
- Friday. Path independence, conservative vector fields.
Wednesday Quiz:
- What is a \(k\)-chain in \(\mathbb{R}^n\)?
- What is the boundary of the standard \(k\)-cube?
- What is the boundary of a \(k\)-surface in \(\mathbb{R}^n\)?
- What is the boundary of a chain?
Homework due Friday:
- HW6.
Week 9
- Monday. Stokes' theorem.
- Tuesday. Review HW.
- Wednesday. Surface integrals.
- Friday. Surface integrals, flux.
Wednesday Quiz:
- Verify Stokes’ theorem in a specific instance.
Homework due Friday:
- HW 7.
Week 10
- Monday. Explanation of flux formula.
- Tuesday. Review HW. Stokes’ theorem. The curl of a vector field.
- Wednesday. Example of Stokes’ theorem. Meaning of the curl. Sage worksheet as pdf, Sage worksheet.
- Friday. The divergence.
Wednesday Quiz:
- Calculate the length of a parametrized curve (perhaps weighted length).
- Calculate the flow of a vector field along a curve.
- What is the gradient of a function? What is a potential function? What is a conservative vector field?
- Show how Stokes’ says that the flow of a conservative vector field along a curve is given by the change in the field’s potential.
Homework due Friday:
- HW 8.
Week 11
- Monday. Divergence. Equivalent conditions for zero divergence.
- Tuesday. Review HW. \(k\)-dimensional surface area.
- Wednesday. Summary of integrals: handout.
- Friday. Maxwell’s equations. Sage worksheet as pdf, Sage worksheet.
Wednesday Quiz:
- Calculate the surface area of a parametrized surface (perhaps weighted surface area).
- Calculate the flux of a vector field through a surface.
- What is the curl of a vector field?
- Show how Stokes’ says that the flow of a vector field along the boundary of a surface is equal to the flux of the vector field’s curl through the surface.
Homework due Friday:
- HW 9.
Week 12
- Monday. Maxwell’s equations. Conservation of charge. Lorentz’ law.
- Tuesday. Review HW. Maxwell’s equations and light.
- Wednesday. Calculating the electric field using symmetry and Gauss’ law. Solutions.
- Friday. Review problems. Solutions.
Wednesday Quiz:
Note: This handout will help.- Explain how Stokes’ theorem implies that the divergence of a vector field measures flux density.
- Explain how Stokes’ theorem implies that the component of the curl of a vector field in a certain direction measures the circulation density of the vector field in a plane perpendicular to that direction.
Homework due Friday:
Week 13
- Monday. Review problems. Solutions.
- Tuesday. Review HW. Alternate form for Maxwell’s equations.
- Wednesday. Review problems. Solutions.
- Friday. Discuss final exam.