MATH 600 Calculus Cheatsheet, Cheat Sheet of Calculus

Download MATH 600 Calculus Cheatsheet and more Cheat Sheet Calculus in PDF only on Docsity! MATH 600 Calculus Cheatsheet This cheatsheet is not, of course, complete. Given a natural number k, a function u : Rm → R is called Ck if u has all possible kth derivatives and the kth derivatives are continuous. A function u : Rm → Rn can be written as u = (u1, . . . , un) where each ui : Rm → R. We say u is Ck if each of its components ui is Ck. A function is C∞ if it is Ck for every natural number k. The support of a continuous function u : Rm → R is the closure of the set {x ∈ Rm|u(x) 6= 0}. We write Ck c = {u ∈ Ck|u has compact support}. The total derivative of u : Rm → Rn at x ∈ Rm, denoted Dux, is the linear map Dux : Rm → Rn which best approximates u near x. In a coordinate system, Dux is represented as a n×m matrix, which consists of the partial derivatives of the component functions. The directional derivative of u : Rm → R at x ∈ Rm in the direction V ∈ Rm is DV ux, the derivative of the function uV (t) = u(x + tV ) at t = 0. In a coordinate system, DV ux = Dux · V . The chain rule: If u : Rm → Rn and v : Rn → Rp are both differentiable, then: D(v ◦ u)x = Dvu(x) ·Dux, where · denotes composition of linear maps (in coordinates, matrix multiplication). Inverse Function Theorem. Let x0 ∈ Rm. Suppose u : Rm → Rm is C1 and has Dux0 invertible. Then there are a neighbourhoods W of x0 and V of z0 = u(x0), such that: • u : W → V is invertible • The inverse v : V →W is as smooth as u is. Implicit Function Theorem. Let (x0, y0) ∈ Rm+k. Suppose u : Rm+k → Rk is C1 and that the k × k- submatrix of Du(x0,y0), ( ∂u ∂y ) is invertible. Then there are neighbourhoods W of (x0, y0) and V of x0, and a map g : V → Rk, such that: • g(x0) = y0 • For any x ∈W , u(x, g(x)) = u(x0, y0) = z0 • For any (x, y) ∈ V , if u(x, y) = z0, then y = g(x). • g is as smooth as u is. 1