Q. 7 Let v聽be a vector in 鈩漬聽and ... [FREE SOLUTION]

Chapter 12: Q. 7 (page 952)

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

Short Answer

Expert verified

Going to assume that limit exists is $D_{u} f (x_{10}, x_{20}, 鈥� 鈥�, x_{n 0}) = {Lim}_{h 鈫� 0} \frac{f (x_{10} + a_{1} 路 h, x_{20} + a_{2} 路 h, 鈥� 鈥� 鈥�, x_{n 0} + a_{n} 路 h) - f (x_{10}, x_{20}, 鈥� 鈥�, x_{n 0})}{h}$

Step by step solution

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.

Equation of v

Directional Derivative of a function of $n$ variables for given vector $v 鈭� {鈩�}^{n}$ :

Let $f (x_{1}, x_{2}, 鈥� 鈥�, x_{n})$ be a function of $n$ variables defined on an open set containing the point $(x_{10}, x_{20}, 鈥� 鈥�, x_{n 0})$ and let $v = (a_{1}, a_{2}, 鈥� 鈥� 鈥�, a_{n})$ be a vector in ${鈩�}^{n}$ .

The directional derivative of $f$ at $(x_{10}, x_{20}, 鈥� 鈥�, x_{n 0})$ in the direction of unit vectoringdenoted by $D_{u} f (x_{10}, x_{20}, 鈥� 鈥�, x_{n 0})$ is given by,

$D_{u} f (x_{10}, x_{20}, 鈥� 鈥�, x_{n 0}) = {Lim}_{h 鈫� 0} \frac{f (x_{10} + a_{1} 路 h, x_{20} + a_{2} 路 h, 鈥� 鈥� 鈥�, x_{n 0} + a_{n} 路 h) - f (x_{10}, x_{20}, 鈥� 鈥�, x_{n 0})}{h}$

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91影视

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

Short Answer

Step by step solution

Introduction.

Equation of v

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91影视

Let vbe a vector in 鈩�nand let fbe a function of nvariables. How would we define the directional derivative of fin the direction of a unit vector u鈭�鈩�vat v?

Short Answer

Step by step solution

Introduction.

Equation of v

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Mechanics Maths

Discrete Mathematics

Calculus

Logic and Functions

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