Chapter 12: Q. 16 (page 953)

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

Short Answer

Expert verified

The function of three variables to be differentiable at the point is $(Δ x, Δ y, Δ z) \to (0, 0, 0)$

Step by step solution

Introduction

Let $f (x, y, z)$ be a three-variable function defined on an open set including the point $(a, b, c)$ , and let $\nabla f (x, y, z) = f (a + ∆ x, b + ∆ y, c + ∆ z) - f (a, b, c)$ be a function of three variables defined on an If the partial derivatives $f x (a, b, c), f y (a, b, c)$ , and $f z (a, b, c)$ exist, the function $f$ is said to be differentiable at $(a, b, c)$ .

Explanation

$\nabla f (x, y, z) = f_{x} (a, b, c) Δ x + f_{y} (a, b, c) Δ y + f_{z} (a, b, c) Δ z + ϵ_{1} Δ x + ϵ_{2} Δ y + ϵ_{3} Δ z$ Where $ϵ_{1}, ϵ_{2}$ and $ϵ_{3}$ are a function of $Δ x, Δ y$ and $Δ z$ , and all goes to zero as $(Δ x, Δ y, Δ z) \to (0, 0, 0)$ .

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

Short Answer

Step by step solution

Introduction

Explanation

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

Short Answer

Step by step solution

Introduction

Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

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Theoretical and Mathematical Physics

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Probability and Statistics

Study anywhere. Anytime. Across all devices.

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