Q. 16 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1 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 $(a, b, c)$.

Step 2 Explanation

$\nabla f(x,y,z)=f_x(a,b,c)\Delta x+f_y(a,b,c)\Delta y+f_z(a,b,c)\Delta z+{ϵ}_1\Delta x+{ϵ}_2\Delta y+{ϵ}_3\Delta z$ Where ${ϵ}_1,{ϵ}_2$ and ${ϵ}_3$ are a function of $\Delta x,\Delta y$ and $\Delta z$, and all goes to zero as $(\Delta x,\Delta y,\Delta z)\to (0,0,0)$.