Math 131



Math 345 –Review for Test 1

Test 1 will consist in 5 problems.

Review the theory for each section, including the proofs for the theorems listed, and homework problems.

Constant & Non-constant Forms

- Vector space, linear map, bilinear map, multilinear map, affine map;

- Calc III Recall:

- Determinants, Jacobian, vectors,

- Dot product, cross-product and oriented area, triple product and volume.

- Differential forms:

- Algebraic definition: multilinear skew symmetric maps;

- Geometric definition: linear combinations of oriented length/area/volume of the corresponding

projections; simplices: (oriented) segment, triangle, tetrahedron;

- Non-constant forms: fields of constant forms;

- Evaluation of constant forms; formulas with determinants

- Pullback of forms: definition & properties (f*(form)(v’s)=form(f(v’s)); (fg)*=g*f*);

- Applications: work and flux;

- Correspondence between vector fields and forms in R, R2 and R3.

Integration

- Parameterized curves & surfaces;

- Simple, double and triple integrals of forms in R2 & R3, using Riemann sums (old definition)

- Line and surface integrals defined as pullbacks (new definition);

- Independence of parameters & pullback of compositions of functions

- Local & global parametrizations: sphere;

- Defining orientations using forms or parametrizations;

- Properties of integrals: orientation, additive, linear functionals;

- Calc III recall: Mean Value Theorem, iterated integrals.

Problems

1) Show that multiplication of real numbers is a bilinear map.

2) Prove that the cross-product of two vectors is a vector orthogonal on each of the two vectors.

3) Evaluate the constant differential form 4dxdy+dydz on the squares determined by the standard basis in R3 (I,j,k).

4) Write the 1-form whose line integral gives the flow of the position vector field F(x,y)=.

Math 345 – Review for Test 2

Test 2 will consist in 5 problems.

Review the theory for each section, including the proofs for the theorems listed, and homework problems.

Ch. 3 Integration and differentiation

- Differentiating forms; d(0-forms) & gradient; d(1-forms) & curl; d(2-forms) & divergence;

- Properties: d linear, d2=0 (w. proof) & geometric interpretation (via Stokes Theorem)

- d2F=0 & equality of mixed partials;

- Higher d2=0: curl(grad)=0, div(curl)=0;

- Closed forms, exact forms; in Rn all closed forms are exact; all k-forms are closed in Rk (top dimension);

- FT in various dimensions (Stokes form)

1) Evaluation 2) Existence of antiderivatives (integrability condition d(form)=0)

- Pullback & differential commute (for functions this is the chain rule);

- Def. exterior differential k-forms; exterior differential;

Ch. 4 Linear algebra

- Linear map & image/rank, kernel; general solution and homogeneous/non-homogeneous solutions;

- Affine manifolds (lines, planes etc.)

- Constant k-forms on Rn; products of forms (algebra of k-forms); basic forms

- Basis Theorem (lexicographic order); algebra of differential forms;

- Matrices & Jacobian;

- Pullback map: linear & associated matrix (“exterior power”); relation with Jacobians

- Implicit Function Theorem: checking conditions & using the conclusion;

- Vector Spaces: def., basis, dimension; vector space of k-forms & their dimension;

- Vector subspaces;

- Linear maps; associated matrix;

- Properties: image, kernel are vector subspaces; dimensions: domain-kernel=image;

- Affine maps, affine manifolds, parametrizations.

Problems (types or samples)

1) Prove the image and kernel of a linear map is a vector subspace.

2) Compute the Jacobian of a map. Compute the pullback of a differential form.

3) Write a differential form in the standard form (basis of forms in lexicographic order).

4) Prove that curl(grad(f))=0.

5) Prove that div(curl(F))=0 (may use the correspondence with differential forms).

6) Prove d2=0 for functions in the plane.

7) Check that the pullback and the differential commute for f(x,y)=(xy,y2) and (=x2dy.

8) Prove that xdy+ydx is a closed form.

9) Prove that xdx+ydy is an exact form.

10) Compute line, surface and volume integrals from differential forms.

Math 345 – Review for Test 3

Test 3 will consist in 5 problems.

Review the theory for each section, including the proofs for the theorems listed, and homework problems.

Ch. 5 Differential calculus

- Partial derivatives, differential functions, Jacobians

- Implicit Function Theorem: determining the rank of a map; solving the equation y=f(x);

- k-forms on n-space: definition, exterior differential, pullback;

- Chain rule: (fg)*=g* f*;

- Implicit differentiation: finding the partial derivatives of the maps (solving dy=f’(x) dx)

- Optimization problems with constraints, method of Lagrange multipliers

- k-dimensional manifolds (defined by equations/parametrization); examples: compact oriented surfaces;

Ch. 6 Integral Calculus

- Measurable regions and k-dimensional volume;

- Change of volume formula

- Integration of k-forms over k-dim. manifolds: decompose + pullback, then iterated integrals;

- Stokes theorem for k-forms in n-space

- Properties of integral:

1) (Integrand) linear, independent of parameters (pullback formula), integration by parts;

2) (Domain) additive, orientation;

3) Extends Riemann integral.

Types of problems:

1) Determine the rank of a map; solve an equation y=f(x);

2) Compute the pullback of a k-form

3) Compute the differential of a k-form;

4) Integrate a k-form (parametrize -> pullback-> iterated integral)

5) Use Stokes Th. to compute an integral (F.T. Calculus)

6) Determine partial derivatives by implicit differentiation;

7) Find max/min using the method of Lagrange multipliers;

8) Find the rate of change of F(x,y) with respect to time if (x,y) moves with given velocity; then write the equation of the tangent line at (x0,y0).

Problems:

1) Discuss the rank of the functions a) (u,v)=f(x,y), u(x,y)=x2-y2, v(x,y)=2xy; b) x=uv, y=u2+v2.

2) Find the pullback of xdy-ydx through f(x,y) from a) above.

3) Compute the gradient and the differential of f(x,y,z)=x2-2y+z2.

4) Find the area of the ellipse x2/a2+y2/b2=1 using a parametrization.

5) Use Stokes Th. to compute [pic] if S is the upper hemisphere of radius 1.

6) Find [pic] and [pic] using the Implicit Function Theorem if y=g(x,z) is defined by z=x2+y2.

7) Maximize V=xyz, if 2z(x+y)+xy=24.

8) Find the rate of change of x2+y2 with respect to time at (x0,y0) if x’=a, y’=b; then write the equation of the tangent line at (x0,y0).

Math 345 – Review for Final Test

(In addition to the reviews for Test 1, 2, 3)

Final Test will consist of 10 problems. 2 pages of definitions, theorems, formulas are allowed (no explicit solutions to homework problems), to be turned in together with the exam.

Review the theory for each section and the homework problems.

Ch. 8 Applications

8.1 Vector calculus in R3

- Line integrals, surface integrals, volume integrals: w. differential forms & vector fields/scalar functions

- Correspondence: differential forms and vector fileds/functions, exterior differentiation and differential operators (grad, curl, div)

- Stokes Theorem (diff. forms) and corresponding theorems in vector field notation

8.2 Elementary differential equations

- D.E., IVP, explicit and implicit solutions

- Def. D.E. with differential forms ( =0 (implicit DE when compared w. dy/dx=f(x,y)) , solutions

- Def. integrating factors (((=dF is exact)

- Solving DE: find an integrating factor (if needed) & then an “antiderivative”

8.3 Harmonic functions and conformal coordinates

- Def. harmonic functions; criterion: solution of Laplace eq.; examples: quadratic forms

- Def. Associated 1-form and orthogonal curves (u -> du -> (= orthogonal form -> (=dv)

- Def. conformal coordinates u,v satisfy Cauchy-Riemann eq. + rank=2;

- Properties: u,v are harmonic functions; coordinate curves are orthogonal; u determines v (using associated orthogonal 1-form); inverse coordinates are conformal; composition of conformal coordinates are conformal coordinates.

8.4 Functions of a complex variable

- Complex numbers, norm (modulus), metric and convergence

- Def. differentiable functions; properties: Cauchy-Riemann equations;

- Def. 1-forms, exterior derivative (0-forms, 1-forms); properties: dzdz=0, all 1-forms are closed;

- Integration of complex 1-forms, Stokes Theorem (k=0: FTC; k=1 Cauchy Theorem), Cauchy formula;

- Properties: any diff. function is analytic; F.T. of Algebra; harmonic functions are analytic.

8.5 Integrability conditions

- Def. systems of DE, solutions of systems of DE, integrability;

- Theorem (Frobenius) System integrable ( (1…(k…d(=0

8.6 Introduction to homology theory - Poincare lemma, k-chains, homology basis

8.7 Flows – description of fluid flowing, conservation equation (d(=(t(+divJ=0)

8.8 Applications to mathematical-physics

- Heat equation & special case (Laplace eq.);

- Potential theory: conservative forces & potential functions;

- Maxewll’s equations (differential forms / duality operator *), d’Alambertian, typical solution (wave)

- Lorentz transformations & special relativity, invariance of Maxwell’s equations.

Types of Problems

1) Computing grad, curl, div;

2) Solving an (implicit) DE ((=0)

3) Prove a function is harmonic; find harmonic functions in a class of functions (quadratic/cubic poly.)

4) Given a harmonic form, write the associated orthogonal form. Integrate it.

5) Given coordinate functions, check they are conformal coordinates.

6) Finding the conformal coordinate partner of a harmonic function (Problem 4 above)

7) Checking a complex function is differentiable (directly or using Cauchy-Riemann eq.)

8) Compute the exterior derivative of a complex function/form

9) Check that a DE system is integrable

10) Check that a 1-form in R2-point is exact

11) Check a function is a solution of the heat equation

12) Compute the differential d(Edt+B) to find 1st 2 Maxwell’s equations from dF=0.

Problems

1) Compute grad(f), f(x,y,z)=2x2+xey+yz.

2) Given F= compute curl(F) (Hint: convert V.F. to 1-form as work, diff. & back to VF as flow)

3) Check xy is a harmonic function.

4) Find all quadratic polynomials in two variables that are harmonic.

5) Given u=xy, find an orthogonal family of curves.

6) Check that u=x2-y2, v=2xy are convormal coordinates in the plane.

7) Prove that f(z)=1/z is a differentiable complex functions.

8) Compute the exterior differential of f(z)=1/z and of the complex 1-form dz.

9) Check that xdy-(xy-y2)dy=0 is integrable (any Adx+Bdy=0 is!)(Hint: compute (d()

10) Prove that (xdx+ydy)/(x2+y2) is exact by computing the value of the pairing on a homology bases.

11) Prove that f(x,y,z)=sin(kr-ct) is a solution of the wave equation (d’Alambertian f=0)

12) Write the vector equations equivalent to d(Edt-B)=0 (curl(E)+(tB=0, div B=0).

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