Multivariable Calculus: Derivatives, Inverse and Implicit Function Theorems
This supplementary multivariable-calculus guide develops total derivatives, partial derivatives, Jacobians and local implicit-function reasoning. Some theorem-level discussion is more advanced than routine admission calculations. It is common mathematical enrichment, not a representation of the target examination’s exact syllabus or difficulty.
Differentiability
f: U ⊆ R^n → R^m is differentiable at a if there is a linear map T with f(a + h) − f(a) − Th = o(||h||). That T is the derivative, represented by the Jacobian matrix of partial derivatives. Existence of all partial derivatives does not imply differentiability; continuous partial derivatives (C¹) do. The chain rule reads D(f∘g)(a) = Df(g(a))Dg(a).
Inverse and implicit theorems
Inverse function theorem: if f is C¹ on an open set, f(a) is defined, and Df(a) is invertible, then f is locally a C¹ bijection onto a neighborhood of f(a) with C¹ inverse, and D(f^{-1})(f(a)) = [Df(a)]^{-1}. Implicit function theorem: if F is C¹, F(a, b) = 0, and the partial Jacobian D_yF(a, b) is invertible, then near a there is a C¹ function g with F(x, g(x)) = 0 and Dg(x) = −[D_yF]^{-1}D_xF. Both conclusions are local; global injectivity is not asserted.
Worked example
Let F(x, y) = x² + y² − 1 and take the point (3/5, 4/5). Then F_y = 2y = 8/5, which is nonzero, so near x = 3/5 there is a C¹ branch y = g(x) with x² + g(x)² = 1, and g'(x) = −F_x/F_y = −x/y = −(3/5)/(4/5) = −3/4. The theorem guarantees a branch near the given point only, not a single formula for all x.
Critical points and Taylor theory
At a critical point, a positive definite Hessian gives a local minimum, a negative definite Hessian a local maximum, and an indefinite Hessian a saddle; semidefinite cases are inconclusive. Schwarz/Clairaut: C² functions have equal mixed partials. Lagrange multipliers apply when the constraint gradient is nonzero, giving ∇f = λ∇g.
Traps
Inferring differentiability from the existence of partials; evaluating the Jacobian at the wrong point; ignoring the invertibility hypothesis; treating local conclusions as global; swapping the x and y roles in the implicit derivative formula; using Lagrange multipliers where the constraint gradient vanishes.