Functions of Several Variables

1. Partial derivatives, differentiability, chain rule

Exam focus: Partials existing ⇏ continuous ⇏ differentiable. The safe implication is: continuous partials ⇒ differentiable. Know xy/(x²+y²) and the equality-of-mixed-partials failure.

Lec-37 Differentiation of Vector Valued Functions

NPTEL · Real Analysis (S. H. Kulkarni, IIT Madras)

The derivative as a linear map — the definition to check in ℝⁿ.

Differentiability in several variables free

The implication chain

continuous partials ⇒ differentiable ⇒ continuous
                                   ⇒ all directional derivatives exist ⇒ partials exist

Every reverse arrow fails.

Function at (0,0)	Partials exist	Continuous	Differentiable
xy $/ (x^{2} + y^{2})$	✓	✗ (limit depends on the path)	✗
$x^{2} y / (x^{4} + y^{2})$	✓, all directional	✗ (limit 1/2 along $y = x^{2})$	✗
$(x^{2} + y^{2}) sin (1/ (x^{2} + y^{2}))$	✓	✓	✓, but partials discontinuous
\|x\| + \|y\|	✗	✓	✗
$x^{3} / (x^{2} + y^{2})$	✓, all directional	✓	✗ (not linear in the direction)

The definition to check: f(a + h) = f(a) + L(h) + o(|h|) for a linear L.

Mixed partials

Clairaut/Schwarz: f_xy = f_yx if both are continuous. Without continuity it fails:

f(x,y) = xy $(x^{2} - y^{2}) / (x^{2} + y^{2}), f (0, 0) = 0$ has f_xy(0,0) = −1 and f_yx(0,0) = 1.

Chain rule and gradients

$- \nabla f$ points in the direction of steepest ascent; the directional derivative is $D_{u} f = \nabla f \cdot u * *$ only when f is differentiable**.

For $x^{3} / (x^{2} + y^{2})$ , directional derivatives exist in every direction but are not linear in u, so f is not differentiable.

Extrema in several variables

Critical point $\nabla f = 0$ ; classify with the Hessian H:

H positive definite ⇒ local min; negative definite ⇒ local max; indefinite ⇒ saddle; singular ⇒ inconclusive.
2 variables: $D = f_{x} x f_{y} y - f_{x} y^{2} . D > 0$ and f_xx > 0 ⇒ min; D > 0, f_xx < 0 ⇒ max; D < 0 ⇒ saddle; D = 0 ⇒ test fails.
Lagrange multipliers for constrained extrema: $\nabla f = λ \nabla g$ .

Key takeaways

Continuous partials is the only cheap sufficient condition for differentiability.
Existence of all directional derivatives does not even give continuity.
Mixed partials commute only under continuity.

2. Inverse and implicit function theorems, extrema

Exam focus: Both theorems need a non-vanishing Jacobian (∂F/∂y ≠ 0 for implicit). Where it vanishes, anything can happen — that is exactly what the questions probe.

Mod-03 Lec-07 Introduction to the Inverse Function Theorem

NPTEL

The statement and the geometric idea: local invertibility wherever the derivative is invertible.

Mod-03 Lec-08 Completion of the Proof of the Inverse Function Theorem

NPTEL

How the proof is finished — worth watching once for where the contraction argument bites.

Mod-04 Lec-10 Introduction to the Implicit Function Theorem

NPTEL

The implicit function theorem as a corollary of the inverse one — the exam's favourite framing.

Lec 32 The Implicit Function Theorem for Functions of Several Variables

NPTEL — IISc Bengaluru

The several-variable statement, with the Jacobian non-vanishing condition made explicit.

Functions of Several Variables

1. Partial derivatives, differentiability, chain rule

Partial derivatives can exist while the function is not differentiableinteractivefree

Differentiability in several variables free

The implication chain

Mixed partials

Chain rule and gradients

Extrema in several variables

Key takeaways

2. Inverse and implicit function theorems, extrema

Implicit function theorem: what to differentiate, and when it says nothinginteractive

Inverse and implicit function theorems

Inverse function theorem