A bird is flying at a speed of 5 m / s in the direction of the vector 4 î+4 ĵ-2 k̂. The tempera

step 1 In this question we have given a rectangular block and it is heated from 0 degrees Celsius to 10

A bird is flying at a speed of 5 m / s in the direction of...

A bird is flying at a speed of $5 \mathrm{~m} / \mathrm{s}$ in the direction of the vector $4 \hat{i}+4 \hat{j}-2 \hat{k}$. The temperature of the region is given by $T=x^{2}+y^{2}-z^{2}$ The rate of increase of temperature per unit time, at the instant it passes through the point $(1,1,2)$ is : (a) $\frac{60}{3}{ }^{\circ} \mathrm{C} / \mathrm{s}$ (b) $3^{\circ} \mathrm{C} / \mathrm{s}$ (c) $18^{\circ} \mathrm{C} / \mathrm{s}$ (d) $4^{\circ} \mathrm{C} / \mathrm{s}$

A bird is flying at a speed of $5 \mathrm{~m} / \mathrm{s}$ in the direction of the vector $4 \hat{i}+4 \hat{j}-2 \hat{k}$. The temperature of the region is given by $T=x^{2}+y^{2}-z^{2}$
The rate of increase of temperature per unit time, at the instant it passes through the point $(1,1,2)$ is :
(a) $\frac{60}{3}{ }^{\circ} \mathrm{C} / \mathrm{s}$
(b) $3^{\circ} \mathrm{C} / \mathrm{s}$
(c) $18^{\circ} \mathrm{C} / \mathrm{s}$
(d) $4^{\circ} \mathrm{C} / \mathrm{s}$

Key Concepts

Gradient

In multivariable calculus, the gradient of a scalar field is a vector that consists of all the partial derivatives of the function. It points in the direction of the greatest rate of increase of the function and its magnitude represents the rate of increase in that direction.

Directional Derivative

The directional derivative measures the rate at which a function changes as you move in a specified direction. It is calculated by taking the dot product of the gradient vector with a unit vector in the direction of interest.

Vector Normalization

Normalization is the process of converting a vector into a unit vector that has a magnitude of 1. This is important when computing directional derivatives because the directional derivative must be measured per unit distance in a particular direction.

Chain Rule in Multivariable Calculus

The multivariable chain rule allows one to compute the derivative of a function along a path by relating the rate of change of the function to the rates of change of its variables. In scenarios involving motion, it connects spatial gradients to time derivatives through the velocity vector.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. It is used to combine the gradient and the unit vector in the desired direction to determine the component of the gradient in that direction, which gives the directional derivative.

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