Partial Derivatives in Engineering Mathematics

Partial Derivatives in Engineering Mathematics: A function is like a machine that takes some input and gives a single output. For example, y = f(x) is a function in ‘x’. Here, we say ‘x’ is the independent variable and ‘y’ is the dependent variable as the value of ‘y’ depends on ‘x’. 

Some examples of functions are:

1. f(x) = x² + 3 is an algebraic function.
2. e^x is the exponential function.
3. sin(x), cos(x), tan(x),…etc. are all trigonometric functions.

Now, all these functions are functions of a single variable, i.e. there is only one independent variable.

Partial derivatives are a fundamental concept in multivariable calculus, often used in engineering mathematics to analyze how functions change when varying one variable while keeping others constant. This is crucial in fluid dynamics, thermodynamics, and structural analysis.

To understand the concept of partial derivative, we must first look at what a function in two variables means. 
Consider a function of the form z = f(x,y) where ‘x’ and ‘y’ are the independent variables and ‘z’ is the dependent variable. This function is called a function in two variables. Similarly, functions of several variables (i.e. with more than 2 independent variables) can also be defined. 

Some examples of multivariable functions or functions of several variables are:

1. f(x,y) = x² +y

2. f(x,y,z) = x-3y+4z

Let us visualize this concept through graph. First we consider a single variable function f(x) = x². 

Unlike functions of a single variable, we cannot visualize multivariable functions as a 2-D graph. For this, we plot it on the 3-D plane. For example, consider the graph of f(x,y) = x²+y²

For functions of several variables we define the limit as follows:

This means, finding the limit of f(x) as ‘x’ approaches ‘a’ and ‘y’ approaches ‘b’. 

Similarly, the definitions of continuity and differentiability can be extended from definitions for single variable functions. 

Recall that the derivative of a function of single variable y=f(x) is defined as : f'(x) = `[Tex]\frac{dy}{dx} [/Tex]

For a function z = f(x,y) of two variables, we define the derivative as : [Tex]\frac{\partial z}{\partial x}[/Tex]

This means calculating the derivative of function ‘z’ with respect to ‘x’ by keeping ‘y’ constant. Similarly, we can calculate the derivative of ‘z’ with respect to ‘y’ by keeping ‘x’ as constant as [Tex]\frac{\partial z}{\partial y}[/Tex]

As we know, for single variable functions, the derivative is computed as the slope of the tangent passing through the curve. Similarly, we can understand the geometric interpretation of a partial derivative of a multivariable function.

Consider a function of two variables, z = f(x,y) on the 3-D plane and let a plane y=b pass through the curve f(x,y). 

Now, we draw another curve f(x,b) lying on z that is perpendicular to the plane y=b. Consider two arbitrary points P,R on this curve and draw the secant passing through these points. 

The slope of this secant is calculated using the first principles as follows :

[Tex]m = \frac{Δz}{Δx} = \frac{f(x+Δx,b)-f(a,b)}{Δx}[/Tex]

As the two points move closer to each other, the difference Δx approaches 0 and we calculate this in the form of the limit : [Tex]\lim_{Δx\to0} \frac{Δz}{Δx} = \frac{f(a+Δx,b)-f(a,b)}{Δx}[/Tex]

This limit is the partial derivative of ‘z’ with respect to ‘x’ by treating ‘y’ as constant i.e. 

[Tex]\frac{\partial z}{\partial x} = \frac{f(a+Δx,b)-f(a,b)}{Δx}[/Tex]

Steps to calculate partial derivative of a given function :

1. Consider z = f(x,y).
2. Compute partial derivative with respect to ‘x’ i.e. [Tex]\frac{\partial z}{\partial x}    [/Tex] by considering ‘y’ as constant and differentiate the function with respect to ‘x’.
3. Compute partial derivative with respect to ‘y’ i.e. [Tex]\frac{\partial z}{\partial y}    [/Tex] by considering ‘x’ as constant and differentiate the function with respect to ‘y’.

Example : [Tex]z = x^2 + y^2 + 3xy[/Tex]

Here, for the given function, we calculate the two partial derivatives as follows : 

Case 1: Differentiating with respect to ‘x’ by treating ‘y’ as constant i.e. [Tex]\frac{\partial z }{\partial x}[/Tex]

Case 2: Differentiating with respect to ‘y’ by treating ‘x’ as constant i.e. [Tex]\frac{\partial z }{\partial y}[/Tex]

Similar to the computation of second-order derivatives for functions of single variables, we can compute the same for functions of several variables.

For an example we consider the same function [Tex]z = x^2 + y^2 + 3xy   [/Tex].

Case 1: We differentiate [Tex]\frac{\partial z}{\partial x}    [/Tex] again with respect to ‘x’

Case 2: We differentiate [Tex]\frac{\partial z}{\partial y}    [/Tex] again with respect to ‘y’

Case 3: We differentiate [Tex]\frac{\partial z}{\partial x}    [/Tex]  again with respect to ‘y’

Case 4: We differentiate [Tex]\frac{\partial z}{\partial y}    [/Tex]  again with respect to ‘x’

People Also Read:

• Partial Derivative: Definition, Formulas and Examples
• Application of Partial Derivative

Partial derivatives are widely used in various engineering disciplines to solve problems involving multiple variables:

• Heat Transfer: Describing the change in temperature distribution over time and space.
• Fluid Dynamics: Analyzing velocity fields and pressure distributions in fluid flows.
• Structural Analysis: Determining stress and strain in materials under load.

What is the physical interpretation of partial derivatives?

Partial derivatives represent the rate of change of a function with respect to one variable, illustrating how a small change in one input affects the output.

How do partial derivatives relate to gradients?

The gradient is a vector of all first-order partial derivatives, indicating the direction of the steepest ascent in a multivariable function.