Chapter 12: Q. 7 (page 952)

Let $v$ be a vector in $ℝ^{n}$ 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 $u \in ℝ^{v}$ at $v ?$

Short Answer

Expert verified

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

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 $v \in ℝ^{n}$ :

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

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

$D_{u} f (x_{10}, x_{20}, \dots \dots, x_{n 0}) = {Lim}_{h \to 0} \frac{f (x_{10} + a_{1} \cdot h, x_{20} + a_{2} \cdot h, \dots \dots \dots, x_{n 0} + a_{n} \cdot h) - f (x_{10}, x_{20}, \dots \dots, x_{n 0})}{h}$

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Let $v$ be a vector in $ℝ^{n}$ 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 $u \in ℝ^{v}$ at $v ?$

Short Answer

Step by step solution

Introduction.

Equation of v

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Let vbe a vector in ℝnand let fbe a function of nvariables. How would we define the directional derivative of fin the direction of a unit vector u∈ℝvat v?

Short Answer

Step by step solution

Introduction.

Equation of v

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Mechanics Maths

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Let $v$ be a vector in $ℝ^{n}$ 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 $u \in ℝ^{v}$ at $v ?$