Q. 7 - Calculus

Question

Let $\mathbf{v}$ be a vector in ${\mathrm{ℝ}}^n$ and let $f$ be a function of $n$ variables. How would we define the directional derivative of $\mathrm{f}$ in the direction of a unit vector $u\in {\mathrm{ℝ}}^v$at $v?$

Step-by-Step Solution

Answer

Going to assume that limit exists is $D_{u}f\left(x_{10},x_{20},\dots \dots ,x_{n0}\right)={Lim}_{h\to 0}\frac{f\left(x_{10}+a_1·h,x_{20}+a_2·h,\dots \dots \dots ,x_{n0}+a_n·h\right)-f\left(x_{10},x_{20},\dots \dots ,x_{n0}\right)}{h}$

Step1: Introduction.

The directional derivative of a multivariable differentiable (scalar) function along a given vector $v$ at a given location $x$intuitively indicates the function's instantaneous rate of change, traveling through $x$ at a velocity described by $v$ in mathematics.

Step2: Equation of v

Directional Derivative of a function of $n$ variables for given vector $v\in {\mathrm{ℝ}}^n$ :

Let $f\left(x_1,x_2,\dots \dots ,x_n\right)$be a function of $n$ variables defined on an open set containing the point$\left(x_{10},x_{20},\dots \dots ,x_{n0}\right)$  and let $v=\left(a_1,a_2,\dots \dots \dots ,a_n\right)$ be a vector in ${\mathrm{ℝ}}^n$.

The directional derivative of $f$at$\left(x_{10},x_{20},\dots \dots ,x_{n0}\right)$ in the direction of unit vectoring  denoted by $D_{u}f\left(x_{10},x_{20},\dots \dots ,x_{n0}\right)$is given by,

$D_{u}f\left(x_{10},x_{20},\dots \dots ,x_{n0}\right)={Lim}_{h\to 0}\frac{f\left(x_{10}+a_1·h,x_{20}+a_2·h,\dots \dots \dots ,x_{n0}+a_n·h\right)-f\left(x_{10},x_{20},\dots \dots ,x_{n0}\right)}{h}$