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Partial derivatives

Probability and statistics, including distributions and hypothesis testing

Vector Calculus and its Applications in Science

Integral and Trigonometric Identities

Matrix Algebra and its applications in Engineering and Physics

Types of Pulleys

Fourier Series and its applications in Signal Processing and Engineering

Partial derivatives are a fundamental concept in multivariable calculus that is widely used in fields such as engineering, physics, and mathematics. A partial derivative is a derivative that is taken with respect to one variable, while all other variables are held constant. This is in contrast to a total derivative, which is the derivative with respect to all variables.

The mathematical definition of a partial derivative of a function, f(x,y), with respect to x, denoted as ∂f/∂x is given by:

Similarly, the partial derivative of f(x,y) with respect to y, denoted as ∂f/∂y, is given by:

Note that, while partial derivatives are taken with respect to a single variable, they are still a measure of how much the function f changes with a small change in x or y. These derivatives provide important information about the local behavior of a function and can be used to optimize, approximate, and solve complex problems in various fields.

The concept of partial derivatives can be generalized to higher dimensions by considering functions of more than two variables. For a function of n variables, f(x1, x2, ..., xn), the partial derivative with respect to x1 is given by:

In addition to their theoretical importance, partial derivatives are widely used in engineering design and construction processes, such as structural analysis and optimization, material design, and fluid dynamics. They also play a crucial role in machine learning and artificial intelligence, where partial derivatives are used to optimize models and learn from data.

Partial derivatives are widely used in various fields of science and engineering, including but not limited to:

Partial derivatives are an essential tool for understanding and solving complex problems in a wide range of fields. They provide important information about the local behavior of functions and are used to optimize, approximate, and solve complex problems. Whether you are a university student or a STEM professional, a strong understanding of partial derivatives is critical for success in your field.

Check out our articles on Vector Calculus, Integrals and Trigonometric Identities, and Differential Equations!

The Fourier series is a mathematical tool widely used in engineering and signal processing to represent periodic signals in terms of a sum of sinusoids. 

It was discovered by Joseph Fourier in the early 19th century and is based on the idea that any periodic signal can be represented as an infinite sum of harmonically related sine and cosine functions. 

The Fourier series is an important tool for analyzing and processing signals in many engineering and physics applications, including filtering, audio and image processing, and communication systems.

Fourier series utilises the orthogonality (that being the 90-degree phase shaft) of sine and cosine functions. By using this principle one can effectively break down any repeating wave into a series of equations that can be solved individually and recombined to obtain an extremely close approximation of the original wave. 

As can be viewed in the above figure, these repeating functions are effectively broken down and reconstructed using a sum of sine and cosine functions. The accuracy of the approximation depends on the number of summations that have taken place. As per the graph, as the number of sine and cosine waves increase (denoted by n), the graph starts to approximate the desired function. It can also be thought that as n approaches infinity, the approximation will be exactly equal to the original function. 

Fourier Series Representation of a Periodic Signal 

A periodic signal f (t) with period T can be represented as a Fourier series as follows:

Where a0, an, bn and n are represented as: 

Try increasing the number of samples and see how the graph changes! (Max: 10)

Number of samples (max 10)

Chart Title

Applications of Fourier Series in Signal Processing and Engineering 

One of the most important applications of the Fourier series is in the representation of periodic signals. By representing a periodic signal as a sum of harmonically related sine and cosine functions, it is possible to analyze and manipulate the signal in a variety of ways. For example, by only keeping the first few terms of the Fourier series, it is possible to simplify and approximate a complex signal, which can be useful in filtering and compression applications.

Another important application of the Fourier series is in the analysis of non-periodic signals. By using the Fourier transform, it is possible to transform a non-periodic signal into the frequency domain, where the signal can be analyzed in terms of its frequency components. This is particularly useful in filtering applications, where it is possible to remove unwanted frequency components from a signal.

The Fourier series is also widely used in audio and image processing, where signals are often represented in terms of a sum of sinusoids. By using the Fourier series, it is possible to analyze and manipulate audio and image signals in terms of their frequency components, which can be useful in a variety of applications, such as noise reduction, compression, and filtering.

The Fourier series is a powerful mathematical tool that is widely used in engineering and signal processing. By representing a periodic signal as a sum of harmonically related sine and cosine functions, it is possible to analyze and manipulate signals in a variety of ways, including filtering, audio and image processing, and communication systems. With its wide range of applications and its simplicity and versatility. The Fourier series is an essential tool for any engineer or physicist working in the fields of signal processing and engineering.

Vector calculus is a branch of mathematics that deals with vector-valued functions, such as vector fields. It plays a crucial role in physics and engineering, as it allows us to describe the relationships between vectors and their derivatives. The fundamental concept of vector calculus is to describe a vector field in terms of its gradient, divergence, and curl. These concepts allow us to describe physical phenomena, such as fluid flow, electromagnetic fields, and the behavior of solids.

The gradient of a scalar field f(x,y,z) is a vector-valued function that describes the rate of change of the scalar field. Mathematically, it is represented as:

The gradient of a scalar field is a measure of the steepness and direction of the field. In physics, 

The divergence of a vector field F(x,y,z) is a scalar-valued function that describes the rate of flow of the vector field. Mathematically, it is represented as:

The divergence of a vector field represents the rate at which the vector field is spreading out or converging. In physics, t

The curl of a vector field F(x,y,z) is a vector-valued function that describes the rate of rotation of the vector field. Mathematically, it is represented as:

The curl of a vector field represents the rate at which the vector field is rotating.

Line integrals in Vector Calculus can refer to either: 


A scalar field line integral is a type of integration that is calculated over a curve in space. Given a continuous function f (x, y, z) and curve C:

Vector field line integrals are a step more complex than scalar field line integrals. It calculates the work done by a vector field along a particular path. A vector field is a mathematical representation of a physical quantity, such as a velocity, force, or electric field, that has magnitude and direction at each point in space.

Let r (t) be a regular parametrisation of the positive-orientated curve C in the set of all 3D vectors (R^3). The line integral of a vector field F along C is given by:

T is the unit tangent vector to C with a positive orientation 

Line integrals are used in various fields of physics, such as mechanics, electromagnetism, and thermodynamics, to calculate physical quantities like work, energy, and heat. For example:

Green's theorem is a result that relates the line integral of a vector field around a simple closed curve to the double integral of the curl of the vector field over the region enclosed by the curve. Green's theorem is a statement about the relationship between the curl of a vector field and the flow of the field through a closed curve. It is named after the British mathematician George Green, who first published the theorem in 1828.

Green's Theorem can be described mathematically as:

Green's theorem is particularly useful in calculating line integrals in complex vector fields.

Stokes' theorem is a result that relates the line integral of a curl of a vector field around a simple closed curve to the surface integral of the curl over a surface bounded by the curve. It is named after the Irish mathematician George Stokes, who first published the theorem in 1845.

Stoke's theorem can be described mathematically as: 

Stokes' theorem is particularly useful in calculating line integrals in complex vector fields. It has applications in:

Gauss' Theorem, also known as Gauss' Divergence Theorem or Ostrogradsky's Theorem, is a result in vector calculus that relates the flow of a vector field through a closed surface to the distribution of its sources and sinks within the volume enclosed by the surface. 

Mathematically, it can be stated as follows:

Gauss' Theorem has a wide range of applications in physics, including fluid dynamics, electromagnetism, and thermodynamics. In particular, it can be used to calculate the total charge enclosed within a closed surface, the total electric flux through a closed surface, and the total heat flow through a closed surface.

Each of these concepts in vector calculus is a vital piece of knowledge for a range of different fields. Whether it's physics, engineering, fluid dynamics, electromagnetism, or even quantum mechanics, these fundamental equations are exceptionally useful in building thorough models of phenomena observed in the universe.

Matrix algebra is a branch of mathematics that deals with matrices, or arrays of numbers, and their operations. It is a fundamental tool for solving linear systems of equations and is widely used in engineering and physics. In this article, we will go over some of the key concepts of matrix algebra and its applications.

A matrix is an array of numbers or variables arranged in rows and columns. A matrix can be represented using bracket notation, with the matrix elements separated by semicolons. For example, the following matrix has two rows and columns:

Matrices can be added, subtracted, and multiplied similarly to scalar numbers. For two given matrices, A and B:

A+B can be represented mathematically as: (This is very intuitive) 

A-B can be represented mathematically as: (This is very intuitive) 

Matrix multiplication is defined as the product of two matrices, with the resulting matrix having the same number of rows as the first matrix and the same number of columns as the second matrix. The resulting matrix elements are found by multiplying elements from each matrix and summing the products. 

A multiplied by B can be represented as: (Not as intuitive) 

The identity matrix is square, with ones on the main diagonal and zeros elsewhere. It is the matrix equivalent of the number 1 for scalars. 

The identity matrix for a matrix of size n x n is given by:

An identity matrix can be used to find the inverse of a matrix. It can also be used to find the eigenvalues and eigenvectors of a matrix. 

Matrices can be inverted. However, the process is not the same as simply inversing each of its elements. A normal inversion of a fraction can be multiplied with its original number to yield the number 1. However, multiplying a matrix by its inverse must yield the identity matrix in the domain of matrices. 

Eigenvalues and eigenvectors are important concepts in matrix algebra. An eigenvalue of a matrix is a scalar corresponding to a linear matrix transformation. An eigenvector of a matrix is a non-zero vector that, when multiplied by the matrix, results in a scalar multiple of the vector. In other words, the eigenvector is stretched or shrunk by the matrix transformation.

Eigenvalues and eigenvectors can be found by solving the characteristic equation of the matrix, which is obtained by using 

denotes the determinant function. The eigenvalues represent the magnitude of the transformation, while the eigenvectors represent the direction of the transformation.

In linear algebra, an eigenvector of a square matrix is a non-zero vector that results in a scalar multiple of itself when multiplied by the matrix. The scalar value is known as the eigenvalue of the eigenvector. 

Mathematically, this can be represented as:

Diagonalization transforms a matrix into a diagonal matrix, where all off-diagonal entries are zero. A matrix is diagonalizable if and only if it has n linearly independent eigenvectors, where n is the number of columns or rows in the matrix. Diagonalization is important because it simplifies the calculation of matrix powers, exponentials, and logarithms, and also provides a basis for understanding the geometry of linear transformations.

A square matrix A is diagonalizable if it can be transformed into a diagonal matrix D using an invertible matrix P:

Matrix algebra is widely used in engineering and physics for solving linear systems of equations, calculating matrix powers, exponentials, and logarithms, and understanding the geometry of linear transformations.

In engineering, matrix algebra is used in control theory, computer graphics, and signal processing. In control theory, matrices are used to model and control systems, including linear time-invariant systems, linear time-varying systems, and nonlinear systems. In computer graphics, matrices are used to transform 3D objects and perform rotations, scaling, and translations. In signal processing, matrices are used to transform signals and perform operations such as filtering, compression, and modulation.

In physics, matrix algebra is used in quantum mechanics, general relativity, and quantum field theory. In quantum mechanics, matrices are used to describe quantum states, operators, and measurements. In general relativity, matrices are used to describe spacetime geometry and objects' motion in gravitational fields. In quantum field theory, matrices are used to describe the behaviour of particles and fields in the presence of interactions.

Matrix algebra is a fundamental tool in engineering and physics, with applications in control theory, computer graphics, signal processing, quantum mechanics, general relativity, and quantum field theory. Understanding eigenvalues and eigenvectors, diagonalization, and matrix operations is essential for solving linear systems of equations, understanding the geometry of linear transformations, and modelling and analyzing systems in engineering and physics.

Probability and statistics play a significant role in many fields, including physics and engineering. These disciplines often require the understanding and analysis of complex data sets and making predictions based on that data. In this article, we will focus on the basic concepts and equations of probability and statistics, including distributions and hypothesis testing, as they are applied in physics and engineering.

A distribution is a representation of the frequency or probability of occurrence of a particular outcome or set of outcomes in a random event. In probability and statistics, several types of distributions are commonly used:

The normal distribution, also known as the Gaussian distribution, is one of the most widely used distributions in physics and engineering. It is a continuous probability distribution with a bell-shaped curve that is symmetrical about its mean. The equation for the normal distribution is given by:

The binomial distribution is used to represent the number of successful outcomes in a fixed number of independent Bernoulli trials. The equation for the binomial distribution is given by:

The Poisson distribution is used to represent the number of events occurring in a fixed interval of time or space. It is often used to model the frequency of rare events. The equation for the Poisson distribution is given by:

The exponential distribution is used to model the time between events in a Poisson process. The equation for the exponential distribution is given by:

Hypothesis testing is a statistical method used to test a claim or hypothesis about a population parameter. In physics and engineering, hypothesis testing is often used to test the validity of a model or to determine if a relationship exists between two variables. 

The basic steps in hypothesis testing are:

The most commonly used test statistic in hypothesis testing is the t-statistic, which is used to test the mean of a normally distributed population. The equation for the t-statistic is given by:

is the probability of observing a test statistic as extreme or more extreme than the one calculated, given that the null hypothesis is true.

If the p-value is less than the significance level:

then the null hypothesis is rejected and the alternative hypothesis is accepted. This means that there is sufficient evidence to support the claim that the population parameter is different from what was stated in the null hypothesis. On the other hand, if the p-value is greater than the significance level then the null hypothesis is not rejected and it is concluded that there is not enough evidence to support the claim.

Hypothesis testing is used in many fields, including physics and engineering, to test the validity of models or to determine relationships between variables. 

For example, in physics, hypothesis testing can be used to determine if a new model for predicting the behavior of a physical system is accurate. 

In engineering, hypothesis testing can be used to determine if there is a relationship between two variables, such as the strength of a material and its composition.

It is important to note that hypothesis testing is only a starting point and additional analysis may be required to confirm the results. It is also important to choose the appropriate test and significance level based on the problem being solved and the data being analyzed.

Here are the top 30 Integral and Trigonometric Identities, along with the latex syntax and any explanatory text for each identity:

These identities are used to convert the sum of sine or cosine into a product, making it easier to integrate.

These identities are used to convert the difference between sine or cosine into a product, making it easier to integrate.

These identities are used to convert between sine and cosine in situations where one is more convenient to use.

These identities are used to simplify expressions and to find the inverse trigonometric functions.

These identities are used to convert between sine and cosine and their respective reciprocals, tangent and cotangent.

Other Hyperbolic Trigonometric Relationships

Then use trigonometric identities to integrate.

The Gamma function, represented as Γ(x), is an extension of the factorial function to real and complex numbers. It is used in many areas of mathematics, including combinatorics, probability theory, and the solution of differential equations. The gamma function has many useful properties, including the ability to evaluate definite integrals and to represent solutions to many special functions, such as the hypergeometric function and the Bessel functions.

Elliptic functions are a class of transcendental functions that are defined on an elliptic curve. These functions are used in many areas of mathematics, including number theory, algebraic geometry, and the solution of differential equations. Elliptic functions have many applications in physics and engineering, including the description of wave propagation, fluid flow, and the calculation of electronic and optical properties.

Pulley systems are an essential component in many mechanical systems and have been used for thousands of years to lift heavy loads and transmit power. In this article, we will explore the different types of pulley systems, their underlying physics, and their various applications.

A pulley is a wheel attached to a shaft or axle that carries a rope, wire or general taut cable across it. It is used to support movement by transferring power, enabling the direction of motion to be changed. 

They can be coupled together to reduce the amount of force required to move or lift a load, at the cost of requiring a longer distance to move the cable through. 

Pulleys are considered to be a part of a group of fundamental mechanical devices known as simple machines.

A fixed pulley system is the simplest type of pulley system, where the pulley is attached to a stationary object such as a wall or a beam. This type of pulley system does not change the direction of the applied force, but it does change the magnitude of the force. The mechanical advantage of a fixed pulley system is equal to the number of ropes supporting the load.

A load with a mass (m1) of 10kg is hoisted up by a person with a force of 1000N directly downwards on a fixed pulley. Determine the acceleration of the system and the force with which the pulley acts upon its axis. Neglect friction, the mass of string and pulley. 

Here we can see all the major forces acting on the system. The applied force (1000N), The tension on the rope (T1 and T2) and the gravitational force exerted by the load (mg).

For our purposes, the m1 is a stand-in for the case where a mass and gravitational force are acting as the pulling force. 

In our system, the magnitudes of the tension on the rope are equal thus simplifying our equations. 

Here's a calculation setup and ready for you to use.

💬 We'd love your feedback on this template! 

 By applying a 1000N force downward, we can accelerate the 10kg mass upwards with an acceleration of 8.06 ms^-2 


A movable pulley system, also known as a block and tackle, is similar to a fixed pulley system but the pulley is attached to the load rather than a stationary object. This type of pulley system changes the direction of the applied force and also increases the magnitude of the force. The mechanical advantage of a movable pulley system is equal to the number of ropes supporting the load multiplied by the number of sections in the pulley system.

A movable pulley system has two masses, M1 = 10kg and M2 = 5kg. Pulley A is fixed, and Pulley B is movable. Assuming the pulleys are massless, the string is taut and a constant tension is maintained in the system, calculate the acceleration of M1. 

If we analyse pulley B, we can recognise that the total tension on the string at both sides is equal to the magnitude of the gravitational force of mass 2. 

Assuming the string is kept taut, M2 will have the same acceleration as pulley B.

Now we must analyse the change in the length of the pulley string sections. 

What's important here is the change in length, not the length of the string itself, a common point of confusion!

Differentiating this change in length twice gives us the resulting accelerations we are after: 

We can calculate the net forces acting on both pulleys using our previous analysis. 

For our movable pulley system with a M1 = 10kg, M2 = 5kg, M1 accelerates downwards at a rate of 6.54 metres/second^2

A compound pulley system is a combination of fixed and movable pulley systems, where multiple pulleys are used to increase the mechanical advantage of the system. This type of pulley system can provide a significant increase in mechanical advantage over a simple fixed or movable pulley system.

Multiple pulley systems are used when a large mechanical advantage is required, such as in cranes and hoists. This type of pulley system can provide an even larger mechanical advantage than a compound pulley system, as the number of sections in the pulley system is increased.

Given the following pulley system, find the force F required to hold a weight W as shown in the below diagram.

It can be observed that at all points on the string, there is a constant tension T. This tension force is equal to the force variable F.

From this diagram, we can determine that the weight force of 100N must equal 5 multiplied by the tension T if the system is to remain in equilibrium.

Let's determine F using our calculation tool.

 Our analysis determined that it would take a 20N force F to maintain equilibrium. Therefore we can ascertain that this pulley system allows for a mechanical advantage of 5 (100/20). 


This demonstrates that for large multiple pulley systems, we can achieve the large mechanical advantages required for machines such as cranes.

A planetary pulley system is a type of pulley system that uses a planetary gear set to transmit power. This type of pulley system is commonly used in power transmission systems and is known for its high mechanical advantage and efficiency.

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