Question

Find the rate of change of temperature (in °C/m) at the point P(2, -1, 3) in the direction toward the point (5, -2, 5).

Answer 1

To find the rate of change of temperature at a specific point in a specific direction, we can use the gradient of temperature. The gradient is a vector that points in the direction of the steepest increase in temperature.

Step-by-step explanation:

To find the rate of change of temperature at a specific point in a specific direction, we can use the gradient of temperature. The gradient is a vector that points in the direction of the steepest increase in temperature. To calculate the gradient, we need to take the partial derivatives of temperature with respect to each coordinate (x, y, z). Let's say the temperature is given by the function T(x, y, z). The gradient vector is given by:

∇T = (∂T/∂x, ∂T/∂y, ∂T/∂z)

To find the rate of change of temperature in the direction toward the point (5, -2, 5), we need to find the unit vector in that direction. Let's call it u. The rate of change of temperature in that direction can be calculated as the dot product of the gradient vector and the unit vector:

Rate of change of temperature = ∇T · u

Let's calculate it step by step.