Question

Zaphod Beeblebrox was in trouble after the infinite improbability drive caused the Heart of Gold, the spaceship Zaphod had stolen when he was President of the Galaxy, to appear between a small insignificant planet and its hot sun. The temperature of the ship's hull is given by \(T(x, y, z)=\) \(e^{-k\left(x^{2}+y^{2}+z^{2}\right)}\) Nivleks. He is currently at \((1,1,1)\), in units of globs, and \(k=2\) globs \(^{-2}\). (Check the Hitchhikers Guide for the current conversion of globs to kilometers and Nivleks to Kelvin.) a. In what direction should he proceed so as to decrease the temperature the quickest? b. If the Heart of Gold travels at \(e^{6}\) globs per second, then how fast will the temperature decrease in the direction of fastest decline?

Short answer

a. The direction of quickest temperature decrease is \((4e^{-2}, 4e^{-2}, 4e^{-2})\) in terms of globs. b. The speed of temperature decrease is \(12e^{4}\) Nivleks per second.

Step-by-step solution

Compute the Gradient of T

The gradient of a function \(f(x, y, z)\) in 3 dimensions is given by \(\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)\). Therefore, the gradient of \(T\) is \(\nabla T = \left( -2kxe^{-kx^2}, -2kye^{-ky^2}, -2kze^{-kz^2}\right)\). Evaluate this at the point \((1, 1, 1)\) to get \(\nabla T(1, 1, 1) = \left(-2ke^{-k}, -2ke^{-k}, -2ke^{-k}\right)\). Substitute \(k = 2\) and compute to get \(\nabla T(1, 1, 1) = \left(-4e^{-2}, -4e^{-2}, -4e^{-2}\right)\).

Determine the Direction of Steepest Descent

The direction of steepest descent is given by the negative of the gradient computed in step 1. Thus, the direction in which Zaphod should proceed is \(-\nabla T(1, 1, 1) = \left(4e^{-2}, 4e^{-2}, 4e^{-2}\right)\).

Compute the Speed of Temperature Decrease

The rate at which the temperature decreases when moving in the direction of steepest descent is given by the scalar product of the gradient and the direction of movement. In this case, since the Heart of Gold travels at \(e^{6}\) globs per second, the direction of movement is scaled by this value. Thus, the rate of decrease is given by \(-\nabla T(1, 1, 1) \cdot \left(e^{6}, e^{6}, e^{6}\right) = 12e^{4}\). This measures the speed of temperature decrease in Nivleks per second.

Key concepts

Gradient of a Function

The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function. In physical terms, if you imagine walking on a terrain, the gradient would point uphill and its magnitude would tell you how steep the hill is.

In mathematical terms for a three-dimensional function like the temperature field given by Zaphod's dilemma, the gradient is a vector consisting of the partial derivatives with respect to each variable:\[\begin{equation}abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)\end{equation}\]
. For the temperature function T(x, y, z), the gradient indicates the direction and rate at which the temperature increases the fastest.

By calculating this, Zaphod can understand which way to move to change the temperature at the highest possible rate. In his case, to escape the heat, he should move in the opposite direction to the gradient, called the direction of steepest descent, which leads to the coolest areas.

Direction of Steepest Descent

The direction of steepest descent is simply the opposite of the gradient vector. If the gradient shows you where the slope rises the most, the direction of steepest descent shows you where it falls the most. For functions like Zaphod's temperature field, which we want to minimize, this direction is of great interest.

In Zaphod's adventure, finding the direction of steepest descent helps him determine the best path to cool down the spaceship. By computing the negative of the gradient at the given point (1, 1, 1), Zaphod finds the direction where the temperature decreases the fastest.The idea is analogous to finding the best way down a hill in the least amount of time; it's generally straight down along the steepest path. For Zaphod, this means navigating the Heart of Gold in the vector direction given by \(-abla T\).Following this path ensures the quickest escape from the scorching effects of the nearby sun.

Scalar Product in Vector Calculus

Now, to quantify the actual rate at which the temperature drops, we use the scalar product, also known as the dot product, in vector calculus.

The scalar product is a mathematical operation that takes two vectors and returns a single number, a scalar. It is given by the formula\[\begin{equation}A \cdot B = |A||B|\cos(\theta),\end{equation}\]where |A| and |B| are the magnitudes of vectors A and B, and \(\theta\) is the angle between them. For Zaphod's case, we calculate the scalar product of the gradient and the velocity vector of the Heart of Gold.This product will tell us the speed of the temperature decrease as the ship moves in the direction of steepest descent. In essence, it combines the direction and magnitude of the two vectors to quantify the change in Zaphod's temperature environment with respect to time, answering part b of the exercise.