Chapter 12: Q.13 (page 953)

What does it mean for a function of two variables, $f (x, y),$ to be differentiable at a point $(a, b)$ ?

Short Answer

Expert verified

Differentiability at the point of function for two variables is $Δ x$ and $∆ y,$ and both goes to zero as $(∆ x, ∆ y) \to (0, 0) .$

Step by step solution

Introduction

Differentiability for function of two variables

$f (x, y)$ be a function of two variables defined on an open set containing the point $(a, b) .$ and let $\nabla f (x, y) = f (a + Δ x, b + Δ y) - f (a, b)$ The function $f$ is said to be differentiable at

$(a, b)$ if the partial derivative $f_{x} (a, b)$ and $f_{y} (a, b)$ both exist and

$\nabla f (x, y) = f_{x} (a, b) Δ x + f_{y} (a, b) Δ y + ϵ_{1} Δ x + ϵ_{2} Δ y$

Solution

Where $ε_{1}$ and $ε_{2}$ are function of $Δ x$ and $Δ y$ , and both goes to zero as $(Δ x, Δ y) \to (0, 0)$ .

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What does it mean for a function of two variables, $f (x, y),$ to be differentiable at a point $(a, b)$ ?

Short Answer

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Introduction

Solution

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What does it mean for a function of two variables, f(x,y),to be differentiable at a point(a,b) ?

Short Answer

Step by step solution

Introduction

Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Decision Maths

Calculus

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

What does it mean for a function of two variables, $f (x, y),$ to be differentiable at a point $(a, b)$ ?