Problem 30 - Advanced Engineering Mathematics

Question

Find a vector that gives the direction in which the given function decreases most rapidly at the indicated point. Find the minimum rate. $$ F(x, y, z)=\ln \frac{x y}{z} ;\left(\frac{1}{2}, \frac{1}{6}, \frac{1}{3}\right) $$

Step-by-Step Solution

Answer

The vector is \((-2, -6, 3)\) and the minimum rate is \(-7\).

Step 1: Compute the Gradient

The gradient of the function, \( F(x, y, z) = \ln \frac{xy}{z} \), is computed by taking partial derivatives. The gradient vector is given by \( abla F = \left( \frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z} \right) \). Compute \( \frac{\partial F}{\partial x} = \frac{1}{x} \), \( \frac{\partial F}{\partial y} = \frac{1}{y} \), and \( \frac{\partial F}{\partial z} = -\frac{1}{z} \). Thus, the gradient is \( abla F = \left( \frac{1}{x}, \frac{1}{y}, -\frac{1}{z} \right) \).

Step 2: Evaluate the Gradient at the Point

Substitute the point \( \left( \frac{1}{2}, \frac{1}{6}, \frac{1}{3} \right) \) into the gradient vector \( abla F = \left( \frac{1}{x}, \frac{1}{y}, -\frac{1}{z} \right) \). This yields \( abla F \left( \frac{1}{2}, \frac{1}{6}, \frac{1}{3} \right) = \left( \frac{1}{\frac{1}{2}}, \frac{1}{\frac{1}{6}}, -\frac{1}{\frac{1}{3}} \right) = (2, 6, -3) \).

Step 3: Determine the Direction of Maximum Decrease

The direction in which the function decreases most rapidly is the opposite direction of the gradient vector. Hence, the direction is \( -abla F = (-2, -6, 3) \).

Step 4: Calculate the Minimum Rate of Decrease

The minimum rate of decrease is the negative of the magnitude of the gradient vector evaluated at the point. The magnitude of \( abla F \) is computed as \( \|abla F\| = \sqrt{2^2 + 6^2 + (-3)^2} = \sqrt{49} = 7 \). Thus, the minimum rate of decrease is \(-7\).

Key Concepts

Partial Derivatives

Partial derivatives are crucial in multivariable calculus and are used to find the rate of change of a function with respect to each variable individually. In essence, they measure how a function changes as one particular variable changes, while keeping others constant.
For instance, in the function given by \( F(x, y, z) = \ln \frac{xy}{z} \), we took the partial derivative with respect to \( x \), treating \( y \) and \( z \) as constants, resulting in \( \frac{1}{x} \).

• Partial derivative with respect to \( y \) is \( \frac{1}{y} \).
• Partial derivative with respect to \( z \) is \( -\frac{1}{z} \).

Finding these derivatives helps construct the gradient, which we'll discuss further, and plays a pivotal role in understanding the behavior of functions in multidimensional spaces.

Gradient Vector

The gradient vector is an essential tool in calculus for vector-valued functions. It combines all the partial derivatives of a function into one vector, which points in the direction of the greatest rate of increase of the function.
For our function \( F(x, y, z) = \ln \frac{xy}{z} \), the gradient is expressed as \( abla F = \left( \frac{1}{x}, \frac{1}{y}, -\frac{1}{z} \right) \).

• The components \( \frac{1}{x}, \frac{1}{y}, \) and \( -\frac{1}{z} \) are derived from the partial derivatives of the function.
• It evaluates to \( (2, 6, -3) \) when you plug in the given point \( \left( \frac{1}{2}, \frac{1}{6}, \frac{1}{3} \right) \).

This gradient vector tells us how the function changes around this particular point, indicating the direction of steepest ascent. To find the path of steepest descent, simply use the negative of this vector, \( -abla F \), which is \( (-2, -6, 3) \) in our case. The magnitude of the gradient provides further context regarding the rate of change.

Function Optimization

Function optimization involves finding the input values that either maximize or minimize a function's output. Gradient descent, a popular method, leverages the gradient vector to find the minimum point of a function by iteratively moving in the direction of the greatest decrease.
The exercise revolves around pinpointing the direction and rate at which our function \( F(x, y, z) = \ln \frac{xy}{z} \) decreases most rapidly.

• By evaluating the gradient vector at a specific point, we ascertain the direction of rapid decrease: the negative gradient, \( (-2, -6, 3) \).
• The rate of this decrease is given by the magnitude of the gradient vector, computed as \( 7 \), resulting in a minimum rate of decrease of \(-7\).

This method helps in efficiently finding the minimum value positions in functions, which is especially useful in various fields like machine learning, economics, and engineering to make well-informed decisions.