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Analysis & Linear Algebra

Functions of Several Variables

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1. Partial derivatives, differentiability, chain rule

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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

In plain words

Partial derivatives look along two axes out of infinitely many, so a function can behave on both and still jump along the diagonal. Differentiable is the stronger demand: locally flat from every direction.

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/(x2+y2)/(x^{2} + y^{2}) ✓ ✗ (limit depends on the path) ✗
x2y/(x4+y2)x^{2}y/(x^{4} + y^{2}) ✓, all directional ✗ (limit 1/2 along y=x2)y = x^{2}) ✗
(x2+y2)sin⁡(1/(x2+y2))(x^{2}+y^{2})\sin(1/(x^{2}+y^{2})) ✓ ✓ ✓, but partials discontinuous
|x| + |y| ✗ ✓ ✗
x3/(x2+y2)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(x2−y2)/(x2+y2),f(0,0)=0(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

−∇f- \nabla{}f points in the direction of steepest ascent; the directional derivative is Duf=∇f⋅uD_uf = \nabla{}f \cdot u only when f is differentiable.

  • For x3/(x2+y2)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 ∇f=0\nabla{}f = 0; classify with the Hessian H:

  • H positive definite ⇒ local min; negative definite ⇒ local max; indefinite ⇒ saddle; singular ⇒ inconclusive.
  • 2 variables: D=fxxfyy−fxy2.D>0D = f_xx f_yy - f_xy^{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: ∇f=λ∇g\nabla{}f = \lambda\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.

Partial derivatives can exist while the function is not differentiableinteractivefree

Step 1 / 4The candidate

Let

f(x,y)=xyx2+y2 for (x,y)≠(0,0),f(0,0)=0.f(x,y) = \frac{xy}{x^2+y^2} \text{ for } (x,y) \neq (0,0), \qquad f(0,0)=0.

Is ff differentiable at the origin?

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

Practice

Part C2025 Decembershow ▾
Let a,b be distinct positive real numbers. Consider the function f:R2→Rf:\mathbb{R}^{2}\to\mathbb{R} given by f(x,y)=(ax+by)2/()^{2}/(ax2+^{2}+by2)^{2}) if (x,y)≠(0,0), and 0 if (x,y)=(0,0). Which of the following statements are necessarily true?
  1. A.lim(x,y)→(0,0) f(x,y) does not exist.✓ Correct
  2. B.The partial derivatives of f at (0,0) do not exist.✓ Correct
  3. C.lim(x→0) f(x,0) = lim(y→0) f(0,y).
  4. D.f is differentiable at (0,0).

The worked solution and the per-option analysis are in Trap Analysis.

Part C2025 Juneshow ▾
Let f:R2→Rf : \mathbb{R}^{2} \to \mathbb{R} be a twice continuously differentiable non-zero function such that f(tx1_{1}, tx2)=t3f(x1,x2)_{2}) = t^{3}f(x_{1}, x_{2}) for all t > 0 and (x1,x2)∈R2(x_{1}, x_{2}) \in \mathbb{R}^{2}. Which of the following statements are necessarily true?
  1. A.3∂f/∂x1(1,1)+3∂f/∂x2(1,1)=f(1,1)3 \partial{}f/\partial{}x_{1}(1, 1) + 3 \partial{}f/\partial{}x_{2}(1, 1) = f(1, 1)
  2. B.∂f/∂x1(1,−1)−∂f/∂x2(1,−1)=3f(1,−1)\partial{}f/\partial{}x_{1}(1, -1) - \partial{}f/\partial{}x_{2}(1, -1) = 3f(1, -1)✓ Correct
  3. C.x12∂2f/∂x12(x1,x2)+x22∂2f/∂x22(x1,x2)+2x1x2∂2f/∂x1∂x2(x1,x2)=6f(x1,x2)x_{1}^{2}\partial^{2}f/\partial{}x_{1}^{2}(x_{1}, x_{2}) + x_{2}^{2}\partial^{2}f/\partial{}x_{2}^{2}(x_{1}, x_{2}) + 2x_{1}x_{2} \partial^{2}f/\partial{}x_{1}\partial{}x_{2}(x_{1}, x_{2}) = 6f(x_{1}, x_{2})✓ Correct
  4. D.x12∂2f/∂x12(x1,x2)+x22∂2f/∂x22(x1,x2)+2x1x2∂2f/∂x1∂x2(x1,x2)=9f(x1,x2)x_{1}^{2}\partial^{2}f/\partial{}x_{1}^{2}(x_{1}, x_{2}) + x_{2}^{2}\partial^{2}f/\partial{}x_{2}^{2}(x_{1}, x_{2}) + 2x_{1}x_{2} \partial^{2}f/\partial{}x_{1}\partial{}x_{2}(x_{1}, x_{2}) = 9f(x_{1}, x_{2})

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let g be a real-valued continuous function on the set {(x,y)∈R2(x, y) \in \mathbb{R}^{2} | x2+y2=1x^{2} + y^{2} = 1} such that g(0, 1) = g(1, 0) = 0 and g(−x, −y) = −g(x, y). Define f:R2→Rf : \mathbb{R}^{2} \to \mathbb{R} by f(x,y)=x2+y2⋅g(x/x2+y2,y/x2+y2)f(x, y) = \sqrt{x^{2} + y^{2}}\cdot{}g(x/\sqrt{x^{2} + y^{2}}, y/\sqrt{x^{2} + y^{2}}) if (x, y) ≠ (0, 0), and f(x, y) = 0 if (x, y) = (0, 0). For each (a,b)∈R2(a, b) \in \mathbb{R}^{2}, define h(a,b):R→Rh_(a,b) : \mathbb{R} \to \mathbb{R} by h_(a,b)(t) = f(ta, tb). Which of the following statements are necessarily true?
  1. A.The function h_(a,b) is differentiable on R\mathbb{R} for each (a,b)∈R2(a, b) \in \mathbb{R}^{2}.✓ Correct
  2. B.There exists (a,b)∈R2(a, b) \in \mathbb{R}^{2} such that h_(a,b) is not differentiable at t = 0.
  3. C.The function f is differentiable at the point (0, 0).
  4. D.If the function f is differentiable at (0, 0) then g is identically zero.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Define f:R2→Rf : \mathbb{R}^{2} \to \mathbb{R} by f(x,y)=yx2+y2/xf(x, y) = y\sqrt{x^{2} + y^{2}}/x if x ≠ 0, and f(x, y) = 0 if x = 0. Which of the following statements are true?
  1. A.∂f/∂x(0,0)\partial{}f/\partial{}x(0, 0) exists✓ Correct
  2. B.∂f/∂y(0,0)\partial{}f/\partial{}y(0, 0) exists✓ Correct
  3. C.f is not continuous at (0, 0)✓ Correct
  4. D.f is not differentiable at (0, 0)✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Consider the function f:R2→Rf : \mathbb{R}^{2} \to \mathbb{R} defined by f(x,y)=x2y/(x4+y2)+e(f(x, y) = x^{2}y/(x^{4} + y^{2}) + e^(xy) for (x, y) ≠ (0, 0), and f(x, y) = 1 for (x, y) = (0, 0). Which of the following statements are true?
  1. A.f is differentiable on R2\mathbb{R}^{2} \ {(0, 0)}.✓ Correct
  2. B.All the directional derivatives of f exist at (0, 0).✓ Correct
  3. C.f is differentiable on R2\mathbb{R}^{2}.
  4. D.f is not continuous at (0, 0).✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
For a positive real number a,aa, \sqrt{a} denotes the positive square root of a. Consider the function f:R2→Rf : \mathbb{R}^{2} \to \mathbb{R} defined by f(x, y) = (x/|x|)x2+y2)\sqrt{x^{2} + y^{2}} for x ≠ 0, and f(x, y) = 0 for x = 0. Which of the following statements are true?
  1. A.f is continuous at (0, 0).✓ Correct
  2. B.The partial derivatives ∂f/∂x\partial{}f/\partial{}x and ∂f/∂y\partial{}f/\partial{}y exist at (0, 0).✓ Correct
  3. C.f is differentiable at (0, 0).
  4. D.f is not differentiable at (0, 0).✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let ‖·‖ denote the Euclidean norm on R2\mathbb{R}^{2} and α≥0\alpha \ge 0. Let f:R2→Rf : \mathbb{R}^{2} \to \mathbb{R} be a function such that |f(x)| ≤ ‖x‖α^\alpha. Which of the following statements are necessarily true?
  1. A.f is differentiable at 0 when α>1\alpha > 1.✓ Correct
  2. B.f is differentiable at 0 when α=1/2\alpha = 1/2.
  3. C.f is continuous at 0 when α=0\alpha = 0.
  4. D.f is continuous at 0 when α>0\alpha > 0.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾
Let p:R2→Rp : \mathbb{R}^{2} \to \mathbb{R} be the function p(x, y) = x. Which of the following statements are true?
  1. A.Let A1=A_{1} = {x2+y2<1x^{2} + y^{2} < 1}. For each γ∈p(A1)\gamma \in p(A_{1}) there exists ε>0\varepsilon > 0 with (γ−ε,γ+ε)⊆p(A1)(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{1}).✓ Correct
  2. B.Let A2=A_{2} = {x2+y2≤1x^{2} + y^{2} \le 1}. For each γ∈p(A2)\gamma \in p(A_{2}) there exists ε>0\varepsilon > 0 with (γ−ε,γ+ε)⊆p(A2)(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{2}).
  3. C.Let A3=A_{3} = {xy = 0}. For each γ∈p(A3)\gamma \in p(A_{3}) there exists ε>0\varepsilon > 0 with (γ−ε,γ+ε)⊆p(A3)(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{3}).✓ Correct
  4. D.Let A4=A_{4} = {xy = 1}. For each γ∈p(A4)\gamma \in p(A_{4}) there exists ε>0\varepsilon > 0 with (γ−ε,γ+ε)⊆p(A4)(\gamma - \varepsilon, \gamma + \varepsilon) \subseteq p(A_{4}).✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾
For a differentiable surjective function f : (0,1) → (0,1), consider F:(0,1)2→(0,1)2,F(x,y)=(f(x),f(y))F : (0,1)^{2} \to (0,1)^{2}, F(x, y) = (f(x), f(y)). If f′(x) ≠ 0 for every x ∈ (0,1), which of the following statements are true?
  1. A.F is injective.✓ Correct
  2. B.f is increasing.
  3. C.For every (x′,y′)∈(0,1)2(x', y') \in (0,1)^{2}, there exists a unique (x,y)∈(0,1)2(x, y) \in (0,1)^{2} such that F(x, y) = (x′, y′).✓ Correct
  4. D.The total derivative DF(x, y) is invertible for all (x,y)∈(0,1)2(x, y) \in (0,1)^{2}.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Decembershow ▾
For real numbers a, b, c, d, e, f, consider the function F:R2→R2F : \mathbb{R}^{2} \to \mathbb{R}^{2} given by F(x, y) = (ax + by + c, dx + ey + f). Which of the following statements are true?
  1. A.F is continuous.✓ Correct
  2. B.F is uniformly continuous.✓ Correct
  3. C.F is differentiable.✓ Correct
  4. D.F has partial derivatives of all orders.✓ Correct

Solution

An affine map is Lipschitz (hence uniformly continuous), differentiable with constant derivative, and all higher partials are zero.

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2. Inverse and implicit function theorems, extrema

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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.

Inverse and implicit function theorems free

In plain words

Both theorems are local, and that single word is the whole content. A map can be reversible around every individual point and still fail to be reversible overall.

Inverse function theorem

f∈C1f \in C^{1} near a, det Df(a) ≠ 0 ⇒ f is a local C1C^{1} diffeomorphism near a, and D(f⁻1)(f(a))=[^{1})(f(a)) = [Df(a)]⁻1^{1}.

Local only. f(x,y) = (eˣcos y, eˣ sin y) has non-zero Jacobian everywhere but is not injective (period 2π2\pi in y). Same phenomenon as eᶻ in complex analysis.

In one variable, f′ ≠ 0 everywhere on an interval does give global injectivity (Darboux forces a constant sign) — the failure is genuinely multivariable.

Implicit function theorem

F(x,y) = 0 with F∈C1,F(a,b)=0F \in C^{1}, F(a,b) = 0 and ∂F/∂y(a,b)≠0\partial{}F/\partial{}y(a,b) \ne 0 ⇒ near a there is a unique C1C^{1} function y = g(x) with F(x, g(x)) = 0, and g′ = −F_x/F_y.

When ∂F/∂y=0\partial{}F/\partial{}y = 0 — the trap

F(x,y) = 0 At (0,0) Solutions y = g(x)
x2−y3x^{2} - y^{3} Fy=−3y2=0F_y = -3y^{2} = 0 exactly one continuous g(x) = x2/3x^{2/3}, not differentiable
x2+y2x^{2} + y^{2} F_y = 0 only the point (0,0); no function
y2−x2y^{2} - x^{2} F_y = 0 two: y = ±x
y3−y−xy^{3} - y - x F_y = −1 at (0,0) theorem applies, unique smooth g

So a vanishing F_y can give zero, one, or many solutions — the theorem gives no information, it does not forbid solutions.

Rank theorem and Lagrange

  • Constant rank ⇒ f looks locally like a projection.
  • Lagrange multipliers are the implicit function theorem in disguise: at a constrained extremum ∇f=λ∇g\nabla{}f = \lambda\nabla{}g, provided ∇g≠0\nabla{}g \ne 0.

Key takeaways

  • Non-zero Jacobian ⇒ local invertibility; global needs more (injectivity is not automatic in Rn)\mathbb{R}^{n}).
  • F_y ≠ 0 is the hypothesis; where it fails, count solutions by hand. −x2=y3- x^{2} = y^{3} is the standard "unique but non-differentiable solution" example.

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

This visual is part of Trap Analysis

A worked implicit derivative, plus the point where the hypothesis fails and the curve really does misbehave.

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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.

Practice

Part C2025 Decembershow ▾
Let F:R3→RF : \mathbb{R}^{3} \to \mathbb{R} be a continuously differentiable function such that F(0,0,0)=−1,∂F/∂x(0,0,0)=2,∂F/∂y(0,0,0)=0F(0,0,0)=-1, \partial{}F/\partial{}x(0,0,0)=2, \partial{}F/\partial{}y(0,0,0)=0, and ∂F/∂z(0,0,0)=3\partial{}F/\partial{}z(0,0,0)=3. For ε>0\varepsilon>0, define Ωε=(−ε,ε)×(−ε,ε)⊂R2\Omega_\varepsilon=(-\varepsilon,\varepsilon)\times(-\varepsilon,\varepsilon)\subset\mathbb{R}^{2}. Which of the following statements are necessarily true?
  1. A.There exist ε>0,δ>0\varepsilon>0, \delta>0 and a continuously differentiable function g:Ωε→(−δ,δ)g:\Omega_\varepsilon\to(-\delta,\delta) such that g(0,0)=0 and F(x,y,g(x,y))=−1 for all (x,y)∈Ωε(x,y)\in\Omega_\varepsilon.✓ Correct
  2. B.There exist ε>0,δ>0\varepsilon>0, \delta>0 and a continuously differentiable function h:Ωε→(−δ,δ)h:\Omega_\varepsilon\to(-\delta,\delta) such that h(0,0)=0 and F(x,h(x,z),z)=−1 for all (x,z)∈Ωε(x,z)\in\Omega_\varepsilon.
  3. C.There exist ε>0,δ>0\varepsilon>0, \delta>0 and continuously differentiable functions k1,k2:(−ε,ε)→(−δ,δ)k_{1},k_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta) such that k1(0)=k2(0)=0k_{1}(0)=k_{2}(0)=0 and F(x,k1(x),k2(x))=−1F(x,k_{1}(x),k_{2}(x))=-1 for all x∈(−ε,ε)x\in(-\varepsilon,\varepsilon).✓ Correct
  4. D.There exist ε>0,δ>0\varepsilon>0, \delta>0 and continuously differentiable functions j1,j2:(−ε,ε)→(−δ,δ)j_{1},j_{2}:(-\varepsilon,\varepsilon)\to(-\delta,\delta) such that j1(0)=j2(0)=0j_{1}(0)=j_{2}(0)=0 and F(j1(z),j2(z),z)=−1F(j_{1}(z),j_{2}(z),z)=-1 for all z∈(−ε,ε)z\in(-\varepsilon,\varepsilon).✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Juneshow ▾
Let f:R2→R3f : \mathbb{R}^{2} \to \mathbb{R}^{3} be a differentiable function such that (Df)(0,0) has rank 2. Write f=(f1,f2,f3)f = (f_{1}, f_{2}, f_{3}). Which of the following statements are necessarily true?
  1. A.f is injective in a neighbourhood of (0, 0)✓ Correct
  2. B.There exists an open neighbourhood U of (0, 0) in R2\mathbb{R}^{2} such that f3f_{3} is a function of f1f_{1} and f2f_{2}
  3. C.f maps an open neighbourhood of (0, 0) in R2\mathbb{R}^{2} onto an open subset of R3\mathbb{R}^{3}
  4. D.(0, 0) is an isolated point of f⁻1(^{1}({f(0,0)})✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part C2023 Juneshow ▾
Consider the function f:R2→Rf : \mathbb{R}^{2} \to \mathbb{R} defined by f(x,y)=x2−y3f(x, y) = x^{2} - y^{3}. Which of the following statements are true?
  1. A.There is no continuous real-valued function g defined on any interval of R\mathbb{R} containing 0 such that f(x, g(x)) = 0.
  2. B.There is exactly one continuous real-valued function g defined on an interval of R\mathbb{R} containing 0 such that f(x, g(x)) = 0.✓ Correct
  3. C.There is exactly one differentiable real-valued function g defined on an interval of R\mathbb{R} containing 0 such that f(x, g(x)) = 0.
  4. D.There are two distinct differentiable real-valued functions g on an interval of R\mathbb{R} containing 0 such that f(x, g(x)) = 0.

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let f(x, y) = (eˣcos y, eˣsin y). Which of the following are true?
  1. A.The Jacobian determinant of f is e2e^{2}ˣ, which is nonzero everywhere.✓ Correct
  2. B.By the Inverse Function Theorem, f is locally invertible near every point of R2\mathbb{R}^{2}.✓ Correct
  3. C.f is injective (globally one-to-one) on R2\mathbb{R}^{2}.
  4. D.The local inverse of f near any point is automatically smooth (C∞)(C^\infty).✓ Correct

Solution

(1) a direct computation gives det⁡J=e2\det J = e^{2}ˣ(cos⁡2y+sin⁡2y)=e2(\cos^{2}y + \sin^{2}y) = e^{2}ˣ. (2) nonzero Jacobian everywhere lets the Inverse Function Theorem apply at every point. (4) the IFT's local inverse is always as smooth as f. (3) is false: f(x,y+2π)=f(x,y)f(x, y + 2\pi) = f(x, y), so f is periodic in y and never globally injective, even though it is locally invertible everywhere — the standard example separating local from global invertibility.

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Part Cshow ▾
Let F(x,y,z)=x2+y2−z2−1F(x, y, z) = x^{2} + y^{2} - z^{2} - 1. Which of the following are true?
  1. A.Near (1, 1, 1), F = 0 implicitly defines z as a smooth function z(x, y), by the Implicit Function Theorem.✓ Correct
  2. B.At (1,1,1),∂z/∂x=1(1, 1, 1), \partial{}z/\partial{}x = 1.✓ Correct
  3. C.At (1,1,1),∂z/∂y=1(1, 1, 1), \partial{}z/\partial{}y = 1.✓ Correct
  4. D.Near (1, 0, 0), F = 0 also implicitly defines z as a smooth function of (x, y).

Solution

∂F/∂z=−2z\partial{}F/\partial{}z = -2z. At (1,1,1) this is −2 ≠ 0, so the IFT applies and ∂z/∂x=−(∂F/∂x)/(∂F/∂z)=x/z=1,∂z/∂y=y/z=1\partial{}z/\partial{}x = -(\partial{}F/\partial{}x)/(\partial{}F/\partial{}z) = x/z = 1, \partial{}z/\partial{}y = y/z = 1, giving (1)–(3). (4) is false: at (1,0,0),∂F/∂z=0(1, 0, 0), \partial{}F/\partial{}z = 0, so the IFT's hypothesis fails — and indeed it must: near x = 1 with y≈0,z2=x2+y2−1y \approx 0, z^{2} = x^{2} + y^{2} - 1 changes sign as x crosses 1, so no real single-valued z(x, y) exists there.

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