Mastering calculus in several variables - Understand partial derivatives, jacobian and multiple integrals

Learn about key concepts like partial derivatives, multiple integrals, and vector calculus, and see how they are used in fields such as physics, engineering, and economics. Master the concepts and take your understanding to the next level with our easy-to-understand explanations and examples.

Table of contents

What is included in a calculus several variables course?

The following topics are typically included in a calculus several variables course

1. Coordinates

Coordinates are sets of numbers describing positions in space, as distances from some reference point. We need as many numbers as the space has dimensions to uniquely determine the location of each point.

2. Vector valued functions

Vector valued functions are mappings from any number of inputs, to vectors which have at least two components.

3. Curves and parametrization

Parametrization refers to the process of expressing relations between variables that depend on each other in terms of independent variables, called parameters. This is useful to describe geometrical objects such as curves and surfaces.

4. Functions of several variables

Functions of several variables has a domain that is multi-dimensional. Hence, a function of $n$ variables takes a set of $n$ numbers as its input.

5. Limits and continuity

Continuity for a function of several variables implies that the limit exists as one and the same value in all directions. That is to say that no irregularities arise from slightly changing one or more of its input variables.

6. Geometry

Geometry is a field of mathematics concerned with spacial objects. It provides ways of formally quantifying intuitive concepts such as distance, shape and size.

7. Partial derivatives

A partial derivative is the analogous of regular derivatives for functions of several variables. In fact the process is just the same, and when taking the partial derivative of a function with respect to one variable, all the others are treated as constants. The partial derivative of a function in $x$ and $y$, with respect to $x$, is denoted as: $$\frac{\partial f(x,y)}{\partial x}$$

8. Gradients

The gradient of a function of several variables is a vector that points in the direction of greatest increase, and its magnitude gives the corresponding rate of change. To form the gradient, we take all the partial derivatives of the function and use these as the vector's components. Usually, the symbol $\nabla$ is used to denote the gradient: $$\nabla f(x,y) = \left[\frac{\partial f(x,y)}{\partial x}, \frac{\partial f(x,y)}{\partial y} \right]$$

9. Differentials

Differentials is the notion of infinitesimal changes in variables. The essence of differential calculus is the relations between such changes, which are obtained through derivatives. A differential is signified by putting a $d$ in front of the variable, and for functions that depend on several independent variables, the relative change is additive: $$df(x,y) = \frac{\partial f(x,y)}{\partial x}dx + \frac{\partial f(x,y)}{\partial y}dy$$

10. Extrema

Extrema of a function occur for inputs where it takes on its largest (maxima) and smallest (minima) values. They can be considered over all space (global extrema), or in a confined region (local extrema).

11. Optimization

Optimization is a branch of mathematics concerned with finding maximum possible improvement. With a function describing the quantity we want to enhance, optimization refers to finding the inputs that correspond to extrema. In practice, all input combinations are not always feasible, and only local extrema may be available. This is referred to as constrained optimization.

12. Implicit functions

Implicit functions relate quantities that depend on each other, without stating a direct causal relationship. In contrast to explicit functions that are formulated in terms of only independent variables, implicit ones are equations containing both dependent and independent variables.

13. Jacobian

The Jacobian of a vector value function is a matrix containing all of the first partial derivatives, inserted in a certain order, which represent the coefficients of a linear approximation of the function. In 2D, the Jacobian looks as follows: $$ J(u,v) = \left[\begin{array}{cc} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{array}\right] $$

14. Double integrals

A double integral considers the integration over two variables simultaneously. This extra dimension makes the integration correspond to calculating the volume under a surface, rather than the area under a curve. Two nested integral signs denote this process: $$I = \int_{y_0}^{y_1} \int_{x_0}^{x_1} f(x,y) \;dx \;dy$$ When evaluating the expression, we perform a regular integration twice; once for each variable, and the respective order does not matter.

15. Triple integrals

Triple integrals are are concerned with three variables of integration. The process of evaluating the integral of three independent variables in sync, i.e. taking the triple integral, can be interpreted as calculating the mass of some density function defined in a region of 3D space. We write it as three nested integrals: $$I = \int_{z_0}^{z_1} \int_{y_0}^{y_1} \int_{x_0}^{x_1} f(x,y,z) \;dx \;dy \;dz$$ The calculation is done just like for double integrals, only with one more integral to evaluate.

16. Taylor approximation

A Taylor approximation of a function expresses it in terms of its value at some reference point, and how it changes around that point. For a function of several variables, the rates of change are given by the partial derivatives of the function, and in principle all partial derivatives are needed to perfectly represent the function. This could result in an infinite sum, referred to as a Taylor series, and a finite truncation is then a Taylor approximation.

17. Vector and scalar fields

The vector field is a region in space where each point is associated with a vector. These can be assigned by a vector valued function that maps coordinates to the components making up such vectors. If a function of coordinates outputs only a single number, the result is instead a scalar field. In other words, a field is nothing but a function of the same number of variables as the dimensionality of the space it lives in, and the number of outputs determines the type of the field.

18. Line and surface integrals

A line integral is a single integral, but in contrast with regular integrals, the curve we're integrating over may stretch across multiple dimensions. Similarly, a surface integral is a double integral over a region that is not confined in only two dimensions. To evaluate line and surface integrals, the trick of parametrization is often employed.

19. Vector calculus

Vector calculus is the extension of differentiation and integration to vector fields. The techniques are primarily employed for vector fields in two and three dimensions, but the theory applies to any number of dimensions.

20. Vector calculus theorems

The most fundamental results of vector calculus are summarized in three theorems: Stoke's theorem, Green's theorem, and the divergence theorem. These properties of vector fields treat the phenomena of flux, curl, and divergence.

What is calculus in several variables?

Calculus in several variables, also known as multivariable calculus, is a branch of mathematics which deals with the analysis of functions with several variables. It is based on the concepts from single variable calculus and extends them to problems involving multiple dimensions and several variables.

Calculus in several variables deals with the properties and behavior of multivariable functions, including partial derivatives, multiple integrals (double integrals, triple integrals) and vector calculus. This branch of mathematics is essential in fields such as physics, engineering and economics, where quantities and system can change with respect to more than one independent variable. It is also used for to model real-world problems and analyze the behavior of multi-dimensional systems.

FAQ

What is a partial derivative?

A partial derivative is a derivative of a multivariable function with respect to one variable, while holding the other variables constant. It represents the rate of change of a function along one direction in the multivariable space.

What is a triple integral?

A triple integral is a type of multiple integral in which the domain of integration is a three-dimensional region in space, and the function being integrated is also a function of three variables.

Triple Integrals are also used to evaluate mass, center of mass, moments of inertia and other quantities for solid bodies or distributions of mass.

What is vector calculus?

Vector calculus is a branch of mathematics that deals with vectors and vector-valued functions. It includes concepts such as vector fields, gradient vectors, divergence, and curl, which allow the calculation and analysis of vector-valued functions in multi-dimensional space. It is widely used in fields such as physics, engineering, and computer graphics.

What is calculus in serveral variables used for? - 6 practical use cases

GPS technology

Coordinates are used pretty much everywhere. For example, when your GPS informs you of your position, it is essentially taking a number of coordinates and translating them to everyday language such as - you are 100 meters from McDonald's on the corner.

Machinelearning

Machine learning is becoming an increasingly important field. It is used pretty much everywhere, by small-scale companies, theoretical physicists and in healthcare. The possibilities are endless! How does machine learning actually work? Machine learning is, just like many other cool techniques, powered by mathematics! And in particular, you need to understand the concept of gradient descent . Basically, it's a method that makes the machine have fewer and fewer errors. So without mathematics, machine learning would have been impossible!

Weather forecast

How bad will hurricane season be next year? No one knows for sure, but it stops us not from guessing.

The weather contains such extraordinarily complicated phenomena that predictions based on current conditions may not anticipate catastrophic events until it is too late to act.

Instead, theories of extrema can be our best chance to get a good feel for future behavior well in advance. This one branch of statistics looks at data recorded from past events and estimates the extremes: the best and worst scenarios.

The method is used for complex questions in the social and natural sciences similar, for example to indicate financial crashes, or damage due to earthquakes, before they have occurred.

Finance

In financial mathematics, the modern portfolio theory is a mathematical method to choose financial assets (shares, bonds, etc) in the best way.

What it really comes down to is solving a optimization problem where you want to pick assets for your portfolio so that:

The expected return is maximized
The risk is minimized

The theory was invented in the fifties by the economist Harry Markovitz. The theory has since received widely used and Markovitz later received the Nobel Prize in Economics for his work.

In physics and engineering

Calculus in several variables is a powerful tool that can be used to model and understand the behavior of physical systems. This powerful mathematical framework can applied to a wide range of phenomena, from the flow of fluids to the behavior of electromagnetic fields.

An example of this is the study of fluid dynamics. Engineers and researchers use multivariable calculus to understand how fluids move and how they are affected by external factors forces. By using multivariable calculus, scientists and engineers can study the fluid flow in complex systems such as the human body, which is important to understand blood flow and other physiological processes. In addition, modeling of the water fluid dynamics is also crucial to understanding ocean currents and climate.

Another example of this is in areas such as technology, where triple integrals are used for evaluation of the electric and magnetic fields, for example for evaluation of electromagnetic energy in a region.

Robotics

In roboticsvector-valued functions are used to describe the motion of robots and other mechanical systems. For example, the position of a robot's arm can be described by a vector-valued function of time, which gives the coordinates of the endpoint of the arm as a function of time. By using vector-valued functions to describe the robot's movement, engineers and researchers can analyze the robot's movements and optimize its performance.

Is calculus in several variables hard?

Multivariable calculus can be considered more challenging than single variable calculus , due to the added complexity of working with multiple variables. However, the concepts and techniques of multivariable calculus build on those of single variable calculus, making the transition to multivariable calculus manageable for those who have a solid understanding of Single variable calculus.

In summary, multivariable calculus is a more complex field than single variable calculus because of the added complexity of working with multiple variables. However, the transition to multivariable calculus is manageable with a solid understanding of single variable calculus, and the concepts and techniques of multivariable calculus build on single variable calculus.

What is the difference between single variable calculus and several variable calculus?

Mathematical analysis is a powerful tool for understanding and analyzing mathematical problems, and it comes in two varieties: single variable calculus and multivariable calculus. Although both forms of analysis share many similarities, there are also some important differences and challenges to be aware of when moving from single variable calculus to multivariable calculus.

One of the biggest differences between the two is the number of variables involved. Single variable calculus deals with functions of a single variable, such as x, y, or t, while multivariable calculus deals with functions of two or more variables, such as x, y, and z. This added complexity of working with multiple variables can make multivariable calculus more challenging than single variable calculus.

Another important difference is the type of problem that can be solved with each form of analysis. Single variable calculus is used to analyze problems involving rates of change and optimization, such as finding the maximum or minimum of a function. Multi variable calculus is used to analyze problems involving multiple variables and multiple dimensions, such as finding the volume of a solid or the force on a surface.

Despite these additional challenges, the concepts and techniques of multivariable calculus build on those of single variable calculus. For example, the derivative of a function, which is a fundamental concept in single variable calculus, is also used in multivariable calculus to study how a function changes when its variables change.

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