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Partial derivatives

Probability and Statistics, including Distributions and Hypothesis Testing

Vector Calculus and its Applications in Science

17 Of The Most Important Equations You'll Ever Learn 

Integral and Trigonometric Relationships

Enhance your mathematical understanding by checking out CalcTree's maths articles.

Maths and Physics Articles - CalcTree

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 formula 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.

Application of partial differential equation

Partial derivatives applications are found 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!

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Unleash the Power of Partial Derivatives. Learn how to compute partial derivatives effectively. Enhance your mathematical skills today!

Master Partial Derivatives: Key Concept in Multivariable Calculus

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 normal distribution formula is given by:

The binomial distribution is used to represent the number of successful outcomes in a fixed number of independent 

 trials. The binomial distribution formula 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 Poisson distribution formula 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 

 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.

Unlock the World of Probability and Statistics: Distributions, Hypothesis Testing, and More. Master concepts for data analysis. Discover now!

Hypothesis Testing - Master Probability distributions - CalcTree

In 2012, mathematician Ian Stewart wrote the book "In Pursuit of the Unknown: 17 Equations That Changed the World", aiming to translate these important equations to the general audience. In his book, he describes equations as being in two distinct groups- the first are equations that show relationships between mathematical quantities and the second one are equations that provide information about unknown quantities. 

In this article, we lay out these 17 equations and provide a summary of each one. We'll explore the theory, importance, history and applications of each equation. Enjoy!

Pythagoras's Theorem (or Pythagorean Theorem) relates the lengths of the sides of a right triangle. The square of the hypotenuse (c) length equals the sum of the squares of the other two sides (a and b). It is fundamental in geometry, physics, and engineering.

It's one of the foundational theorems in mathematics, used in various fields, from geometry to physics, providing a fundamental relationship between the lengths of the sides of a right triangle.

Attributed to the ancient Greek mathematician Pythagoras, this theorem has been used for millennia. It was known to Babylonians and Indians well before Pythagoras, but he and his followers may have been the first to prove it.

The Pythagorean Theorem is mostly used in architecture, construction, navigation and surveying [2]. 

After more detail on Pythagoras? Check out our detailed write-up and explanation 

Logarithms are the inverse of exponential functions. For a given base b, the logarithm function 🚨logbx🚨 gives the power to which b must be raised to produce x. It's vital in solving exponential equations.

Logarithms are used extensively in computer science, engineering, and economics, often for complexity analysis and growth modelling.

Invented independently by John Napier and Joost Bürgi in the early 17th century, logarithms dramatically simplified calculations before calculators, especially in astronomy and navigation.

Logarithms vastly reduce the time required in multiplying numbers with multiple digits [3] and are used in surveying, celestial navigation and calculation of noise [4]. Specifically, logarithms led to efficient methods for calculating astronomical phenomena and measure of time radioactivity. 

Calculus deals with the concepts of derivatives and integrals. The derivative represents the slope of the tangent line to the function at a point, giving the function's rate of change. Integrals provide the accumulation of quantities.

It's fundamental in physics, engineering, economics, and other fields for modelling change and motion.

The foundations of calculus were laid independently and simultaneously by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. It led to a bitter dispute over priority.

Calculus has many uses in a variety of industries, mostly for optimization, summation and trend prediction. In business, it used in maximizing profits and minimizing costs [5]. It is also used in various engineering fields like fluid mechanics and determination of frictional forces[6]. 

Newton's law of universal gravitation states that every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

It's the foundation for classical mechanics, explaining phenomena from an apple's falling to planets' motion.

Formulated by Isaac Newton in 1687 in his work "Mathematical Principles of Natural Philosophy," this law revolutionized the understanding of motion and allowed the prediction of planetary movements.

Newton's Law of Universal Gravitation explains the gravitational forces between two bodies [7] by approximating the gravitational strength between two objects [8]. It is used in discovering new planets and stars and predicting tides. It is also used in finding the masses and predicting the path of astronomical bodies [9]. 

Learn more about orbital mechanics and calculate orbital velocity 

The wave equation describes the propagation of waves, such as sound or light. It relates the second spatial and temporal partial derivatives of the wave function.

It's essential in physics, particularly optics and acoustics, for understanding wave behaviour.

Jean d'Alembert derived the one-dimensional wave equation in 1746. This fundamental equation has since become crucial in studying physical phenomena involving waves.

The wave equation is used in various phenomena, including gravitational, light and sound waves and is also used to study the oscillations of string in string theory [7]. It can also help us understand string and fluid surface movement [10]. 

The imaginary unit i is defined as the square root of -1. It forms the basis for complex numbers, extending real numbers to accommodate values like square roots of negative numbers.

Complex numbers are central to advanced mathematics, physics, engineering, and signal processing.

Imaginary numbers were first coined in the 17th century. The concept was met with resistance but eventually became a fundamental part of complex analysis and engineering.

Imaginary numbers are used in signal processing, radar, or any parameter that relies on using the sine and cosine wave [11]. 

Euler's polyhedron formula relates the number of vertices (V), edges (E), and faces (F) of a convex polyhedron. It is a topological invariant.

It's fundamental in geometry and topology, often used in network analysis and computer graphics.

Discovered by Leonhard Euler in the 18th century, this formula united geometry and topology and laid the foundation for graph theory.

A notable example of the application of Euler's Formula is in computer chip designing, to make sure that the number of intersections in the system are as low as possible [12]. 

The normal distribution is a probability distribution that is symmetric about the mean μ and characterized by its standard deviation σ. It describes many natural phenomena.

It's widely used in statistics, finance, and social sciences to model random variables.

The normal distribution is often called the Gaussian distribution, developed by Carl Friedrich Gauss in the early 19th century. It's fundamental in statistics and probability theory.

Applications of the normal distribution include quality control, sales forecasting and tasks that require statistical inference and data analysis [13]. 

Introduction In 2012, mathematician Ian Stewart wrote the book "In Pursuit of the Unknown: 17 Equations That Changed the World", aiming to translate t

17 Of The Most Important Equations You'll Ever Learn

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 

 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. 

 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 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.

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 

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.

Unleash the potential of vector calculus in science. Elevate your understanding with advanced mathematical concepts. Explore now!

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.

Master the Pythagorean Identity, trigonometric relationships, and more to enhance your math skills. Read more to learn about them!