The temperature of a point in space is given by T=(x^2+y^2-z^2) . A mosquito located at (1,1,2)

step 1 In this question it is told that the length of cylinder on heating increases by 2 % then we have

The temperature of a point in space is given by...

The temperature of a point in space is given by $T=\left(x^{2}+y^{2}-z^{2}\right) .$ A mosquito located at $(1,1,2)$ desires to fly in such a direction that it will get heat as soon as possible. Then unit vector is : (b) $-\frac{2}{3} \hat{1}-\frac{2}{3} \hat{j}+\frac{1}{3}$ (i) $\frac{2}{3} \hat{i}-\frac{2}{3} \hat{j}-\frac{1}{3}$ (d) none of these

The temperature of a point in space is given by $T=\left(x^{2}+y^{2}-z^{2}\right) .$ A mosquito located at $(1,1,2)$ desires to fly in such a direction that it will get heat as soon as possible. Then unit vector is :
(b) $-\frac{2}{3} \hat{1}-\frac{2}{3} \hat{j}+\frac{1}{3}$
(i) $\frac{2}{3} \hat{i}-\frac{2}{3} \hat{j}-\frac{1}{3}$
(d) none of these

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Key Concepts

Gradient

The gradient of a scalar field is a vector composed of the partial derivatives with respect to each variable. It points in the direction of the greatest rate of increase of the function, and its magnitude gives the rate of increase. In problems involving temperature, the gradient shows the direction in which temperature increases most rapidly.

Directional Derivative

The directional derivative measures the rate at which a scalar function changes in a specific direction. It is calculated by taking the dot product of the gradient vector and a unit vector in the desired direction, allowing one to determine how fast the temperature or any other scalar field is changing along that path.

Unit Vector and Normalization

A unit vector is any vector with a magnitude of one, used to denote a direction without any scaling. Normalization is the process of dividing a vector by its magnitude to convert it into a unit vector. This concept is essential when computing directional derivatives to ensure that only the direction, not the magnitude of the movement, is considered.

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