Partial Derivative

What is a Partial Derivative?

To calculate a partial derivative, you treat it essentially the same as a normal derivative, with a slight difference:

• Any variable besides the one you are differentiating with respect to must be treated as a constant.
• A function's partial derivative is usually written as \(\frac{\partial f}{\partial x}\) or \(\frac{\partial f}{\partial y}\), where f is the function and x and y are the variables.

Relation of Partial Derivative to Engineering Mathematics

Engineers often deal with multi-variable functions, where more than one input affects the output. In these cases, partial derivatives become absolutely imperative. They allow you to understand how the function behaves with respect to a single variable, even when there could be multiple variables in play. For example, in structural engineering, a construction's stress or displacement might be expressed as a function of both the load and the temperature. With a partial derivative, you can determine how much the stress changes for each unit change in the load, leaving the temperature constant and vice versa. A partial derivative can also be used to find an optimal solution for a problem in engineering. In such cases, you want to minimize or maximize a function, and this is where methods like gradient descent comes into play, which are based on partial derivatives. The table below gives a brief overview of how these methods are employed in different engineering fields. Remember, this is just the tip of the iceberg when it comes to the application of partial derivatives in engineering. The concept is used in numerous other contexts and situations across various engineering fields.

Learning Through Examples: Partial Derivative Examples

Understanding partial derivatives becomes simpler when we delve into examples. Observing how it's employed in hypothetical scenarios and real-world engineering problems can help solidify your knowledge and application of the concept.

Understanding Simple Partial Derivative Examples

Let's start by taking a couple of simple mathematical functions and calculating their partial derivatives. Keep in mind that when you differentiate with respect to one variable, all other variables are considered constants. Consider the function \(f(x, y) = x^2 y + y^3\). To find the partial derivative with respect to \(x\), treat \(y\) as constant. So, \(\frac{\partial f}{\partial x}= 2x*y\). In case of differentiating with respect to \(y\) (while treating \(x\) as constant), we use the power rule of differentiation, hence, \(\frac{\partial f}{\partial y} = x^2 + 3y^2\). Now, for a somewhat more elaborate function, \(f(x, y, z) = x^3 y^2 z + 3x yz^2 + z^3\), the partial derivative with respect to \(x\) is \(\frac{\partial f}{\partial x} = 3x^2 y^2 z + 3yz^2\). Similarly, we compute the partial derivative with respect to \(y\) and \(z\) considering other variables as constants. Therefore, \(\frac{\partial f}{\partial y} = 2x^3 y z + 3x z^2\) and \(\frac{\partial f}{\partial z} = x^3 y^2 + 6x yz + 3z^2\). Working through such examples is a critical part of mastering the concept of partial derivatives.

Partial Derivative Examples in Real World Engineering Problems

Moving onto real-world engineering problems, partial derivatives play a essential role in solving complex multi-variable issues. For instance, you're working as a Mechanical Engineer tasked with modelling the efficiency, \(\eta\), of a heat engine. This efficiency depends on variables such as the engine's temperature (\(T\)), pressure (\(P\)), and the fuel's caloric value (\(C\)). The existing model provides you with this function: \[ \eta(T, P, C) = 0.5T^2P + 0.3TPC + 0.2PC^2 \] To find the rate of change of efficiency with respect to temperature (while keeping pressure and caloric value constant), you would take the partial derivative of \(\eta\) with respect to \(T\). Here, \(P\) and \(C\) are treated as constants. Hence, \(\frac{\partial \eta}{\partial T} = TP + 0.3PC\) Similarly, by taking the partial derivative with respect to \(P\) and \(C\), while treating the others as constants, you can find the rate of traction of the efficiency due to those parameters. Another instance can be seen during the design process in Civil Engineering, where a skyscraper's stability \(S\) is dependent on the wind speed \(W\), building material grade \(G\), and the height \(H\) of the skyscraper. Suppose you have already derived a function that describes this relationship: \[ S(W, G, H) = 0.4W^2G + 0.6WG^2H \] In this scenario, partial derivatives would be used to understand the rate at which the skyscraper's stability changes when wind speed, material grade or building height is varied. During the design process, finding out how much variation in the individual factors affect the stability of the structure can be essential to create a safe and reliable construction. Therefore, as we see from these examples, the use of partial derivatives is vital in dealing with many complex, practical engineering problems. The concept truly stands as a cornerstone of engineering mathematics.

Extending Your Knowledge: Partial Derivative Chain Rule

As you explore more about partial derivatives, you'll encounter a key rule frequently used in calculus--The Chain Rule. This rule, when applied to partial derivatives, has significant application in solving complex multiple-variable problems.

Introduction to the Partial Derivative Chain Rule

The Chain Rule has a long-standing reputation in calculus, dealing with composite functions. So, what exactly is this rule? However, when it comes to multiple-variable functions, the Chain Rule takes on a slightly different form which is known as the Partial Derivative Chain Rule. Mathematically, if \(z\) is a function of \(n\) independent variables \(x_1, x_2, x_3, ..., x_n\), where each \(x_i\) is a function of \(m\) independent variables \(y_1, y_2, y_3, ..., y_m\), the Partial Derivative Chain Rule states that: \[ \frac{\partial z}{\partial y_j} = \frac{\partial z}{\partial x_1}\frac{\partial x_1}{\partial y_j} + \frac{\partial z}{\partial x_2}\frac{\partial x_2}{\partial y_j} + \cdots + \frac{\partial z}{\partial x_n}\frac{\partial x_n}{\partial y_j} \] For \(j = 1, 2, 3, ..., m\). Think of the Chain Rule as a tool to dissect a complex multi-variable function into simpler bits, examine how each part behaves, and then combine the results to get a comprehensive understanding of the original function's behaviour.

Application of the Chain Rule in Partial Derivative

Once you understand the theory, using the Partial Derivative Chain Rule with actual functions becomes more intuitive. It provides a structured method for dealing with compound functions and simplifies complex variables and derivatives. Let's consider a classic engineering problem: heat transfer. Suppose the amount of heat transferred, \(Q\), depends on the difference in temperature (\(T\)) between two bodies and the contact time (\(t\)). The temperature difference itself could again be a function of the initial temperatures of both bodies, \(T_A\) and \(T_B\). In this case, you can represent the problem as: \[ Q(T, t) = Tt^2 \] and \[ T(T_A, T_B) = T_A - T_B \] The question might arise of how \(Q\) changes with \(T_A\) or \(T_B\). Here's where you use the Partial Derivative Chain Rule. Using the chain rule, \[ \frac{\partial Q}{\partial T_A} = \frac{\partial Q}{\partial T}\frac{\partial T}{\partial T_A} \] and \[ \frac{\partial Q}{\partial T_B} = \frac{\partial Q}{\partial T}\frac{\partial T}{\partial T_B} \] By performing these calculations, you will evaluate how the heat transfer changes with respect to \(T_A\) and \(T_B\). So, a small increase in the initial temperature of body A will increase heat transfer at a rate proportional to \(t^2\), while an increase in \(T_B\) will decrease heat transfer at the same rate. Wherever there's a relationship between quantities that indirectly influences another, such as the temperature example above, you'll find that the Partial Derivative Chain Rule can simplify and clearly outline the problem, leading you closer to a solution. It gives you a more significant insight into how the function behaves when more variables are included. In a nutshell, whether it's in fluid mechanics, heat transfer, electrical circuits, or any other engineering field, the chain rule of partial derivatives significantly simplifies the resolution of multi-variable problems.

Diving Deeper: Partial Derivative Examples

To deepen your understanding of partial derivatives, it's highly beneficial to examine more complex examples. By doing this, you'll build the necessary competency to tackle more intricate mathematical and engineering problems that involve calculus of several variables.

The Basics of Partial Derivative Examples

To illustrate the concept of a partial derivative, let's evaluate it for the function \(f(x, y) = x^2 y + y^3\). In essence, the partial derivative of a function of two variables with respect to one of its variables is the derivative of the function treated as a function of only that variable. The other variable is treated as a constant. When we are differentiating \(f(x, y)\) with respect to \(x\), we treat \(y\) as a constant. As a result, the partial derivative of \(f(x, y)\) with respect to \(x\), denoted as \(\frac{\partial f}{\partial x}\), gives us \(2x*y\). On the other hand, when differentiating \(f(x, y)\) with respect to \(y\), we treat \(x\) as a constant. Hence, the resultant \(\frac{\partial f}{\partial y}\) is \(x^2 + 3y^2\). Let's now look at a more complex multi-variable function, \(f(x, y, z) = x^3 y^2 z + 3x yz^2 + z^3\), and the workings of its partial derivatives. When differentiating with respect to \(x\), while treating both \(y\) and \(z\) as constants, we get \(\frac{\partial f}{\partial x} = 3x^2 y^2 z + 3yz^2\). Similarly, by treating the other variables as constants, the partial derivatives with respect to \(y\) and \(z\) are \(\frac{\partial f}{\partial y} = 2x^3 y z + 3x z^2\) and \(\frac{\partial f}{\partial z} = x^3 y^2 + 6x yz + 3z^2\) respectively. As these examples show, the key to finding partial derivatives boils down to your ability to differentiate basic functions while treating the other variables as constants.

Partial Derivative Examples Application in Engineering Mathematics

Partial derivatives find extensive application in the domain of engineering. They help establish how a dependent variable changes with respect to specific independent variables while holding others constant. One common application lies within Thermodynamics. Suppose you are an engineer and you have a heat engine. The efficiency, \(\eta\), of this engine could depend on several variables, such as the engine's temperature (\(T\)), pressure (\(P\)), and the specific heat capacity (\(C\)) of the fuel used. Imagine that you have an efficiency function such as: \[ \eta(T, P, C) = 0.5T^2P + 0.3TPC + 0.2PC^2 \] To understand how changes in \(T\), \(P\), and \(C\) impact the engine's efficiency, you can calculate the respective partial derivatives of \(\eta\) with respect to these variables, while treating the others as constants. Similarly, in structural engineering, the stability \(S\) of a skyscraper could depend on factors like the wind speed \(W\), the building material grade \(G\), and the height \(H\) of the skyscraper. If you have a function that describes this relationship as \(S(W, G, H) = 0.4W^2G + 0.6WG^2H\), you could use partial derivatives to find out the impact of changing wind speeds, material grade, and building height on the stability of the structure. By treating the other variables as constants, you can isolate the effect of one variable on the overall stability. Therefore, partial derivatives provide a fundamental tool in extracting clarity from the complexity. It's an invaluable skill in engineering mathematics, as it enables you to discern the impact of specific independent variables on a dependent variable, an understanding critical in the design and analysis of effective engineering solutions.

Advanced Learning: Second Order Partial Derivative

Progressing deeper into the realm of partial derivatives, you'll encounter the Second Order Partial Derivative, an aspect that provides profound insights into the nature of multivariable functions.

Know-all about Second Order Partial Derivative

In the study of multivariable calculus, second order partial derivatives play a vital role. They essentially measure how first order partial derivatives change concerning other variables, providing deeper information about the shape, concavity, convexity and extrema of the function in question. The first order derivative of a multivariable function \(f\) with respect to \(x\), denoted as \(\frac{\partial f}{\partial x}\), provides the rate of change of the function with changing \(x\) while keeping all other variables constant. This notion extends to the second order derivative, denoted \(\frac{\partial^2 f}{\partial x^2}\), which measures the rate at which the first order derivative changes with changes in \(x\). There are two types of second order partial derivatives:

• Pure second order derivatives: These involve taking the derivative twice with respect to the same variable. For instance, \(\frac{\partial^2 f}{\partial x^2}\) and \(\frac{\partial^2 f}{\partial y^2}\).
• Mixed second order derivatives: This involves taking the derivative first with respect to one variable and then to another. For instance, \(\frac{\partial^2 f}{\partial x \partial y}\) or \(\frac{\partial^2 f}{\partial y \partial x}\).

The order of differentiation matters in the case of mixed partial derivatives. However, under most standard conditions encountered in engineering, thanks to a theorem known as Clairaut's theorem, the mixed second order derivatives \(\frac{\partial^2 f}{\partial x \partial y}\) and \(\frac{\partial^2 f}{\partial y \partial x}\) are equal. Consider a simple example function \(f(x, y) = x^2y - y^3.\) Its mixed second order derivatives are computed as follows: - First take the derivative of \(f(x, y)\) with respect to \(x\) to get \(\frac{\partial f}{\partial x} = 2xy\). - Next, take the derivative of \(\frac{\partial f}{\partial x}\) with respect to \(y\) to get \(\frac{\partial^2 f}{\partial y \partial x} = 2x\). The process can be repeated with reversed variables and will prove the equality of mixed second derivatives as per Clairaut's theorem.

First derivative of f(x, y) with respect to y is 
∂f/∂y = x² - 3y². 
Next, the derivative of ∂f/∂y with respect to x is 
∂²f/∂x∂y = 2x. 

Engineerings Finest: Second Order Partial Derivative in Action

Integral to understanding the intricacies of engineering problems, second order partial derivatives are essential tools in the engineer's mathematical toolbox. This concept is often applied in the analysis and modelling of physical phenomena. In many branches of engineering, it's common to use second order derivatives, especially in solving differential equations. In thermodynamics, second order partial derivatives are typically used to derive certain material properties' relationships. Furthermore, in fluid dynamics and electrical engineering, the second order derivatives aid in solving flow and circuit equations respectively. Consider a fluid flowing in a pipe with pressure \(P\) and velocity \(v\). Suppose the pressure at any point depends on the radius \(r\) and time \(t\), given by the relationship \(P(r, t) = re^{-rt}\). The second order partial derivative \(\frac{\partial^2 P}{\partial r^2}\) can inform us about the pressure distribution's concavity with respect to the pipe's radius, a vital insight when designing efficient fluid systems. Ultimately, the application of second order partial derivatives in engineering extends from the design to analytical phases, and even to the optimisation of systems and processes. It's through these second order relationships that you gain insight into dependencies and variations between variables, which is crucial in predicting and controlling system responses. Remember, the more strides you take into this fascinating world of multivariable calculus and its applications in engineering, the more evident it becomes that these concepts, including second order partial derivatives, are fundamental pieces in understanding the many-layered puzzle of engineering problems.