Chapter 12: Q. 69 (page 954)

Let $(a, b)$ be a point in the domain of the function of two variables, role="math" localid="1650475276808" $f (x, y)$ , and $u$ be a unit vector for which $D_{u} f (a, b)$ exists. Prove that $D_{- u} f (a, b) = - D_{u} f (a, b) .$

Short Answer

Expert verified

Hence proved is $= - D_{u} f (a, b)$

Step by step solution

Variable function.

Let $(x, y)$ be a two-variable function and $u = ⟨ α, β ⟩$ , be a unit vector for $D_{- u} f (a, b)$ The objective is to prove that

$D_{- u} f (a, b) = - D_{u} f (a, b)$

The direction derivative of a function $f (x, y)$ at $(x_{0}, y_{0})$ in the direction of $u = ⟨ α, β ⟩$ is given by

$D_{u} f (x_{0}, y_{0}) = \lim_{h \to 0} \frac{f (x_{0} + α h, y_{0} + β h) - f (x_{0}, y_{0})}{h}$

Thus,

D_{u} f (a, b) = \lim_{h \to 0} \frac{f (a + α h, b + β h) - f (a, b)}{h}

(1)

Find the Equations.

For $- u = ⟨ - α, - β ⟩$

$D_{- u} f (a, b) = \lim_{η \to 0} \frac{f (a - α η, b - β η) - f (a, b)}{η}$

$= \lim_{η \to 0} \frac{f (a + α (- η), b + β (- η)) - f (a, b)}{η}$

Let $h = - η$ then

$D_{- u} f (a, b) = \lim_{h \to 0} \frac{f (a + α h, b + β h) - f (a, b)}{- h}$

$= - D_{u} f (a, b)$ from $(1)$

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Let $(a, b)$ be a point in the domain of the function of two variables, role="math" localid="1650475276808" $f (x, y)$ , and $u$ be a unit vector for which $D_{u} f (a, b)$ exists. Prove that $D_{- u} f (a, b) = - D_{u} f (a, b) .$

Short Answer

Step by step solution

Variable function.

Find the Equations.

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Let (a,b) be a point in the domain of the function of two variables, role="math" localid="1650475276808" f(x,y), and u be a unit vector for which Duf(a,b) exists. Prove thatD-uf(a,b)=-Duf(a,b).

Short Answer

Step by step solution

Variable function.

Find the Equations.

One App. One Place for Learning.

Most popular questions from this chapter

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Let $(a, b)$ be a point in the domain of the function of two variables, role="math" localid="1650475276808" $f (x, y)$ , and $u$ be a unit vector for which $D_{u} f (a, b)$ exists. Prove that $D_{- u} f (a, b) = - D_{u} f (a, b) .$