Problem 3 - Advanced Engineering Mathematics

Question

In Problems, 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: Introduction to Gradient Calculation

The gradient of a function is a vector that consists of the partial derivatives of the function with respect to each variable involved. For the function given, we need to find the partial derivatives with respect to each of the variables: \( x \), \( y \), and \( z \).

Step 2: Compute Partial Derivative with respect to x

To find the partial derivative of \( F(x, y, z) \) with respect to \( x \), keep \( y \) and \( z \) treated as constants. The partial derivative is: \[ \frac{\partial F}{\partial x} = \frac{\partial}{\partial x} \left( \frac{x y^2}{z^3} \right) = \frac{y^2}{z^3} \].

Step 3: Compute Partial Derivative with respect to y

To find the partial derivative of \( F(x, y, z) \) with respect to \( y \), keep \( x \) and \( z \) treated as constants. The partial derivative is: \[ \frac{\partial F}{\partial y} = \frac{\partial}{\partial y} \left( \frac{x y^2}{z^3} \right) = \frac{2xy}{z^3} \].

Step 4: Compute Partial Derivative with respect to z

To find the partial derivative of \( F(x, y, z) \) with respect to \( z \), keep \( x \) and \( y \) treated as constants. The partial derivative is: \[ \frac{\partial F}{\partial z} = \frac{\partial}{\partial z} \left( \frac{x y^2}{z^3} \right) = -\frac{3xy^2}{z^4} \].

Step 5: Combine Partial Derivatives into Gradient

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

Key Concepts

Partial Derivatives

To understand and compute the gradient of a multivariable function, we first need to explore the concept of partial derivatives. A partial derivative measures how a function changes as one of the variables changes, keeping the other variables constant. This is crucial when dealing with functions of more than one variable.For the function given, \( F(x, y, z) = \frac{x y^2}{z^3} \), we have three variables: \( x \), \( y \), and \( z \).

• Partial derivative with respect to \( x \): Treat \( y \) and \( z \) as constants. The derivative \( \frac{\partial F}{\partial x} \) shows how \( F \) changes as \( x \) changes. Here, it simplifies to \( \frac{y^2}{z^3} \).


• Partial derivative with respect to \( y \): Treat \( x \) and \( z \) as constants. The derivative \( \frac{\partial F}{\partial y} \) reflects the change in \( F \) when \( y \) is varied, leading to \( \frac{2xy}{z^3} \).


• Partial derivative with respect to \( z \): Keeping \( x \) and \( y \) constant, \( \frac{\partial F}{\partial z} \) evaluates how \( F \) changes with \( z \), resulting in \( -\frac{3xy^2}{z^4} \).

By calculating these derivatives, we can together form the gradient vector, which captures the direction and rate of the steepest ascent at a given point in the function.

Vector Calculus

Vector calculus plays an essential role in understanding the behavior of functions of multiple variables. It extends the concepts of calculus from functions of a single variable to multivariable. A central concept is the gradient.The gradient of a scalar function like \( F(x, y, z) = \frac{x y^2}{z^3} \), is represented as a vector \( abla F(x, y, z) \) consisting of all calculated partial derivatives:\[ abla F(x, y, z) = \left( \frac{y^2}{z^3}, \frac{2xy}{z^3}, -\frac{3xy^2}{z^4} \right) \]

• Direction: The gradient direction points in the direction of the greatest rate of increase of the function. It acts as a navigational tool indicating where the function is climbing most rapidly.


• Magnitude: Besides direction, the magnitude (length) of the gradient vector tells us how steep the function is ascending in that direction. This is crucial in optimization tasks where we want to find maximum values.

Vector calculus principles such as these offer tools to analyze dynamic systems in physics, engineering, or any field involving changes across more than one dimension.

Multivariable Calculus

The field of multivariable calculus extends single-variable calculus concepts to higher dimensions. This branch studies functions with more than one input variable, crucial for understanding systems in a multidimensional space.In the context of this function, \( F(x, y, z) \), it involves three variables that represent 3-dimensional space. Calculating derivatives in this setting is more complex yet follows foundational principles: holding other variables constant while differentiating concerning one.

• Gradient: A prime example in multivariable calculus, it presents crucial information about the rate and direction of change in space. In practical applications, this can indicate the slope of a hill in any direction.


• Visualization: With multiple inputs and outputs, imagine the surface or curve taking shape. Understanding partial derivatives helps us visualize changes along this surface, enhancing our comprehension of the 3D structure impact.

The significant contribution of multivariable calculus is its comprehensive ability to model real-world scenarios involving varying elements, enabling us to predict and analyze complex systems effectively.