Application of Partial Derivatives in Engineering Mathematics

Application of Partial Derivatives: Partial derivatives can be used to find the maximum and minimum value (if they exist) of a two-variable function. We try to locate a stationary point with zero slope and then trace maximum and minimum values near it. The practical application of maxima/minima is to maximize profit for a given curve or minimize losses.

Let f(x,y) be a real-valued function and let (pt, pt’) be the interior points in the domain of f(x,y) then,

• pt, pt’ is called a point of local maxima if there is an h > 0 such that f(pt, pt’) ≥f(x,y), for all x in (pt – h, pt’ + h), x≠a The value f(pt, pt’) is called the local maximum value of f(x,y).
• pt, pt’ is called a point of local minima if there is an h < 0 such that f(pt, pt’) ≥f(x,y), for all x in (pt – h, pt’ + h), x≠a The value f(pt, pt’) is called the local minimum value of f(x,y).

Partial derivatives are extensively used in engineering to model and solve problems involving multiple variables. These derivatives help in understanding how a system changes with respect to one variable while keeping others constant, providing essential insights into the behavior of physical systems.

1. Find the values of x and y using f_xx=0 and f_yy=0  [NOTE: f_xx and f_yy are the partial double derivatives of the function with respect to x and y respectively.]
2. The obtained result will be considered as stationary/turning points for the curve.
3. Create 3 new variables r,t, and s.
4. Find the values of r,t and s using r=f_xx, t=f_yy, s=f_xy
5. If (rt-s²)|_{(stationary pts)}>0  (Maxima/Minima) exists
6. If (rt-s²)|_{(stationary pts)} <0 (No Maxima/Minima)/(Saddle point)
7. If r=f_xx>0 (Minima) 
8. If r=f_xx<0 (Maxima)

Example-1 :

The function f(x,y)=x²y−3xy+2y+x has                

• (a) No local extremum
• (b) One local minimum but no local maximum
• (c) One local maximum but no local minimum
• (d) One local minimum and one local maximum

Explanation :

Answer: A

r=∂2f/∂x2=2y
s=∂2f/∂x∂y=2x−3
t=∂2f/∂y2=0

Since, rt−s²≤0, (if rt-s²< 0 then we have no maxima or minima, if = 0 then we can’t say anything).

Maxima will exist when rt−s²>0 and r<0.

Minima will exist when rt−s²>0 and r>0.

As rt−s² is never greater than 0 so we have no local extremum.

Example-2 :

Find the local minima of the function f(x , y) = 2x² + 2xy + 2y² – 6x

fx(x,y) = 4x + 2y - 6=0    (1)
fy(x,y) = 2x + 4y=0        (2)

On solving (1) and (2) we get,

x=2,y=-1
r=∂2f/∂x2=4
s=∂2f/∂x∂y=2
t=∂2f/∂y2=4
rt−s2=12

As rt−s²>0 and r>0. Thus, (2,-1) is the point of local minima.

Example-3 :

Find the maxima/minima of  f(x , y) = x²+y² + 6x +12

fx(x,y) = 2x+6=0     (1)
fy(x,y) = 2y=0       (2)

On solving (1) and (2) we get,

x=-3,y=0
r=∂2f/∂x2=2
s=∂2f/∂x∂y=0
t=∂2f/∂y2=2

As rt−s²>0 and r>0. Thus, (-3,0) is the point of local minima.

1. Heat Transfer
   • Fourier’s Law: In heat conduction, the temperature distribution within a solid can be analyzed using partial derivatives. Fourier’s Law uses partial derivatives to describe the rate of heat transfer through a material.
2. Fluid Dynamics
   • Navier-Stokes Equations: These equations describe the motion of fluid substances and are fundamental in predicting weather patterns, designing aircraft, and understanding ocean currents. They use partial derivatives to account for changes in velocity and pressure in the fluid.
3. Structural Analysis
   • Stress and Strain Analysis: Engineers use partial derivatives to calculate the stress and strain on materials. This analysis is crucial for designing structures that can withstand various forces and loads.
4. Electromagnetics
   • Maxwell’s Equations: These equations use partial derivatives to describe how electric and magnetic fields propagate and interact. They are essential in the design of electrical and communication systems.
5. Optimization Problems
   • Maximizing Efficiency: Engineers often use partial derivatives to optimize functions representing cost, efficiency, or other performance measures. By finding the critical points and using second-order partial derivatives, they can determine local maxima and minima.

What are partial derivatives used for in engineering?

Partial derivatives are used to model and solve problems involving multiple variables, such as in heat transfer, fluid dynamics, structural analysis, electromagnetics, optimization, and control systems.

Why are partial derivatives important in fluid dynamics?

Partial derivatives help describe the behavior of fluid substances, allowing engineers to analyze and predict changes in velocity, pressure, and other fluid properties using the Navier-Stokes equations.

How do partial derivatives apply to structural analysis?

In structural analysis, partial derivatives are used to calculate stress and strain on materials, ensuring that structures can withstand various forces and loads without failing.

What is the role of partial derivatives in optimization problems?

Partial derivatives are used to find the critical points of functions representing cost, efficiency, or other performance measures, enabling engineers to optimize these functions.