Question

2. The maximum value of the directional derivative takes place in the direction of ∇Φ and its
magnitude is-
(a) ∣ ∇Φ ∣
(b) ∣ ∇2Φ ∣
(C) ∣ ∇Φ |2

(d) ∇Φ​

Answer 1

Answer:

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Explanation:

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Answer 2

Answer:

The magnitude of maximum directional derivative of a function at a point P is ${|\vec\bigtriangledown \phi|}$ .

Explanation:

• In vector analysis, slope is termed as a gradient, i.e., a vector operator denoted by $\bigtriangledown$.
• If $\phi=f(x,~y,~z)$ is a real function of x, y and z, then $\vec \bigtriangledown f(x,~y,~z)$ or $grad~ f(x,~y,~z)$ is called the gradient of $f$.
• And is given by

        $\vec \bigtriangledown f(x,~y,~z)=f_x(x,~y,~z)\hat x+f_y(x,~y,~z)\hat y+f_z(x,~y,~z)\hat z$

• In a simplified form

         $\vec \bigtriangledown \phi=\frac{\partial \phi}{\partial x} \hat x+\frac{\partial \phi}{\partial y}\hat y+\frac{\partial \phi}{\partial z}\hat z$

• The orientation in which the directional derivative has the largest value is the direction of $\vec \bigtriangledown \phi$ .
• That is the maximum directional derivative of a function $f$ at a given point P is along the same direction of the gradient vector of $f$.
• Which can be written as

        $\hat u=\frac{\vec \bigtriangledown \phi}{|\vec \bigtriangledown \phi|}$

• And ${|\vec\bigtriangledown \phi|}$ is the magnitude of that directional derivative.
• Hence, the magnitude of the maximum directional derivative of a function at a point P is ${|\vec\bigtriangledown \phi|}$ .

So, the correct answer is option (a).