MATH 51Review Session Notes (Sun, 07 Dec 08)1 Vector Calculus1.1 Continuity and LimitsExample Spring03Q1 Consider the function f : R2→ R2given byf(x, y) =(xy(x2+y2)2if (x, y) 6= (0, 0)0 if (x, y) = (0, 0)(a) Do ∂f/∂x and ∂f /∂y exist at the origin? If yes, compute them; if not,explain why.Concept: the definition of partial derivatives.(b) Is the function f continuous at the origin? Explain.A function is not continous at a point a if the value of the function ap-proaches different values as we approach a along different paths (in this case,try approaching the origin via y = x and either of the axes).(c) Is the function f differentiable? Explain.Concept: if a function is differentiable at a p oint a, then it must be contin-uous at a.1.2 Using the Chain RuleExample Given two differentiable functions f, g : R2→ R2such thatD(f ◦ g) =ex00 ey, g(x, y) := (x, ey),find Df(x, ey).Solution. The chain rule says D(f ◦ g)(z) = Df(g(z))Dg(z). So Df(x, ey) =Df(g(x, y)) = (D(f ◦ g))Dg−1.1.3 Directional De rivative, GradientThe directional derivative of a function f : X ⊂ Rn→ R at the point a ∈ X,along the direction of the unit vector v ∈ Rn, is given byDvf(a) := limh→0f(a + hv) − f(a)h= ∇f(a) · v when f is differentiable at aThe value of a function f : X ⊂ Rn→ R increases the most quickly along thedirection of the gradient vector ∇f.1Example Fall00Q16 Let f : R2→ R be defined by f (x, y) = xy2− x3. (a)What is the direction of fastest decrease of f at (1, 1)? Express it as a unitvector. (b) What is the directional derivative of f at the point (1, 2) in thedirection toward the point (4, 3)?Example Let f : X ⊆ R2→ R be differentiable at a ∈ X. Let u :=(1, 0), v := (0, 1). Supp os e Duf(a) = 0 = Dvf(a). Is it possible that forsome unit vector w ∈ R2, Dwf(a) 6= 0? Why or why not? (No. RecallDwf(a) = ∇f(a) · w, and note u, v form a basis for R2)And the gradient vector ∇f(a) at a is perpendicular to the level set passingthrough a, namely {x ∈ X : f (x) = f (a)}.Example (Fall01Q11(a)) Consider the surface defined byx3+ xyz + z3= 3Find the equation of the tangent plane to the surface at the point (1, 1, 1).1.4 Differentials and Taylor PolynomialsExample Given f(x, y) := ex+y. Find the equation of the tangent plane tothe graph of f at a := (0, 0). Find the first order Taylor polynomial of f at a.Solution. Recall the equation for the tangent plane at a isz − f(a) =∂f∂x(a)∆x +∂f∂y(a)∆ywhere ∆x = (x − 0), ∆y = (y − 0).If we write h :=xy−00to denote the ”change” from our reference pointa, then the above equation can be rewritten asz = f(a) + ∇f(a) · hOr, because Df = ∇f, we can write it asz = f(a) + Df(a)h(so that the dot product is replaced by a matrix multiplication in the secondterm).Note that the last expression is exactly the first-order Taylor polynomial.Two remarks:1. If we want to approximate, say, f(0.2, 0.3), then we can say: the point(0.2, 0.3) is close to the point (0, 0), which has coordinates that are arith-metically easier to work with. So we look at the tangent plane (or first-order Taylor polynomial of f at (0, 0)), and use it to approximate thefunction f . (So put in x = 0.2, y = 0.3 in the tangent plane equation, orthe first-order Taylor polynomial, to approximate f(0.2, 0.3).)22. The second-order Taylor polynomial for f at a would involve a furtherterm,12hTHf(a)h. Here Hf(a) is a symmetric matrix (when f is differ-entiable). When Hf(a) is positive definite (i.e. the (1, 1)-entry and thedeterminant are both > 0), the term12hTHf(a)h would always be posi-tive. Draw a picture to illustrate that f has a local minimum at a in thiscase. And when Hf(a) is negative definite, (i.e. the (1, 1)-entry is < 0 andthe determinant is > 0), the term12hTHf(a)h is always negative. Drawanother picture to illustrate that we get a local maximum in this case.1.5 Extrema and Lagrange Multipliers(We’ll skip this, since these are the most recent topics.)2 Linear Algebra2.1 Dot product, cross product• Cauchy-Schwarz Ineq: |v · w| ≤ kvk · kwk.• Triangle Ineq: kv + wk ≤ kvk + kwk.• v · w = |v| · |w| cos(θ)• two nonzero vectors are perpendicular iff v · w = 0.• the cross product v × w of two vectors in R3is orthogonal to both v andw, i.e. v · (v × w) = 0, etc.• kv × wk = kvkkwk sin(θ)• the area of the parallelogram formed by v and w is kv × wk. What aboutthe area of the triangle formed by v and w?Example Fall01Q1(b,c) (b) Find the angle between−141and2−21. (c)Find the area of the triangle with vertices (0, 0, 0), (−1, 4, 1) and (2, −2, 1). Usethe determinant formulation of cross product.2.2 Solutions to Systems of Linear equationsA system of linear equations: has either no solutions, a unique solution, or infi-nitely many solutions. The solution set of a homogeneous system is a subspaceof Rn(if there are n variables involved), and is the nullspace of a matrix (thecoefficient matrix for the system).The solution set of an inhomogeneous system is the translation of the solutionset to the corresponding homogeneous system, by a particular solution.3Example Fall05Q8(b) Consider the system of equations given by3x + 2y − z = ax + 7y − 5z = b5x + 16y − 11z = cFor which values of a, b, c do e s this s ystem of equations have solutions?2.3 Null Space, Column Space, Basis, Dimension, Rank,Nullity, Linear TransformationGood to know (even if you don’t remember how ot prove it): If n > m, thenany set of n vectors in Rmare linearly dependent.How to find a basis for N(A): use the fact N(A) = N(rref(A)). Iden-tify the free variables by looking at rref(A), hence write the solution set to(rref(A))x = 0 in parametric form, with the free variables as the parameters; abasis for N(A) can then be extracted from the parametric form. (section 11 oftext).How to find a basis for C(A): the columns in rref(A) containing pivotscorrespond to the columns in A that form a basis for C(A).Example (Fall04Q1(a,b)) Consider the matricesA =0 0 1 11 3 1 22 6 1 3and R = rref(A) =1 3 0 10 0 1 10 0 0 0(a) Find a basis for the column space of A. (b) Find a basis for the nullspace ofR. (use N(A) = N(rref(A)): very important). (c) Given that A1111=2712,find all solutions to Ax =2712.Example Fall00Q3(b-g) Suppose A a 5 × 5 matrix withrref (A) =1 0 −1 4 00 1 2 3 00 0 0 0 10 0 0 0 00 0