Problem 3 - Advanced Engineering Mathematics

Question

Compute the gradient for the given function. $$ F(x, y, z)=\frac{x y^{2}}{z^{3}} $$

Step-by-Step Solution

Answer

The gradient is \( \left( \frac{y^2}{z^3}, \frac{2xy}{z^3}, -\frac{3xy^2}{z^4} \right) \).

Step 1: Understand the Problem

We need to find the gradient of the function \( F(x, y, z) = \frac{x y^2}{z^3} \). The gradient of a function is a vector composed of its partial derivatives with respect to each variable.

Step 2: Calculate Partial Derivative with Respect to x

To find \( \frac{\partial F}{\partial x} \), treat \( y \) and \( z \) as constants and differentiate with respect to \( x \). The derivative is: \( \frac{\partial F}{\partial x} = \frac{y^2}{z^3} \).

Step 3: Calculate Partial Derivative with Respect to y

To find \( \frac{\partial F}{\partial y} \), treat \( x \) and \( z \) as constants and differentiate with respect to \( y \). Use the power rule: \( \frac{\partial F}{\partial y} = \frac{2xy}{z^3} \).

Step 4: Calculate Partial Derivative with Respect to z

To find \( \frac{\partial F}{\partial z} \), treat \( x \) and \( y \) as constants and differentiate with respect to \( z \). Use the quotient rule: \( \frac{\partial F}{\partial z} = -\frac{3xy^2}{z^4} \).

Step 5: Write the Gradient Vector

The gradient of the function \( F(x, y, z) \) is the vector of its partial derivatives: \( abla F = \left( \frac{y^2}{z^3}, \frac{2xy}{z^3}, -\frac{3xy^2}{z^4} \right) \).

Key Concepts

Partial Derivatives

Partial derivatives are essential in analyzing functions with multiple variables. They help us understand how a function changes as one variable changes, while keeping the others constant. In simple terms, when we find the partial derivative of a function with respect to a variable, we are observing the rate at which the function value changes as only that particular variable changes. For instance, in the original exercise, we have the function \( F(x, y, z) = \frac{x y^2}{z^3} \). Here, to find \( \frac{\partial F}{\partial x} \), we treat \( y \) and \( z \) as constants during differentiation. This helps isolate the effect of \( x \) on \( F \). Similarly, when finding \( \frac{\partial F}{\partial y} \) and \( \frac{\partial F}{\partial z} \), different sets of variables are held constant.

• \( \frac{\partial F}{\partial x} = \frac{y^2}{z^3} \)
• \( \frac{\partial F}{\partial y} = \frac{2xy}{z^3} \)
• \( \frac{\partial F}{\partial z} = -\frac{3xy^2}{z^4} \)

Learning to calculate partial derivatives is crucial as they form the building blocks for more advanced topics like the gradient and optimization of multivariable functions.

Multivariable Calculus

Multivariable calculus extends the concepts we learn in single-variable calculus to functions with more than one input. Such functions are prevalent in physics, engineering, economics, and other fields where systems depend on multiple factors. A distinctive feature of multivariable calculus is its ability to handle and analyze these complex functions.

The gradient is an important concept in multivariable calculus. It combines the partial derivatives into a vector that points in the direction of the steepest ascent of the function, giving us multi-dimensional insights into the behavior of the function. For any given multivariable function, computing the gradient involves:

• Finding all partial derivatives with respect to each independent variable.
• Combining these derivatives into a vector form.

For the function in our exercise, the gradient \( abla F \) is the vector \( \left( \frac{y^2}{z^3}, \frac{2xy}{z^3}, -\frac{3xy^2}{z^4} \right) \). This vector helps us in determining how the function changes in different directions within the space determined by \( x, y, \) and \( z \).

Vector Calculus

Vector calculus provides a framework for differentiating and integrating vector fields—functions that have a vector as their output. It greatly extends the power of calculus to analyze physical phenomena, like electromagnetic fields or fluid flow, which inherently involve vectors. In our context, vector calculus deals with how vectors vary and is key to solving problems related to curves, surfaces, and volumes in 3D space.

The gradient is a classic example of vector calculus in action. When we say that \( abla F = \left( \frac{y^2}{z^3}, \frac{2xy}{z^3}, -\frac{3xy^2}{z^4} \right) \), we are describing a gradient vector field in which each point \( (x, y, z) \) in the space has an associated vector pointing in the direction where the function \( F \) increases most swiftly. This utility highlights how vector calculus helps in understanding the directional derivatives, maxima and minima of multivariable functions, and the paths often traversed by physical entities being studied. The gradient effectively acts like a map indicating the easiest route of change and direction for an object's parameterized movement in three-dimensional spaces.