# Integration in Vector Fields - Mathematics

Integration in Vector Fields - Mathematics

## VECTOR CALCULUS

Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus is the foundation stone on which a vast amount of applied mathematics is based. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis.

The basic objects in vector calculus are scalar fields (scalar-valued functions) and vector fields (vector-valued functions). These are then combined or transformed under various operations, and integrated. In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields except that they change sign under an orientation-reversing map: for example, the curl of a vector field is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction.

## LEEDS MATHS TUITION I mentioned vector fields in a previous post in the context of differential equations and over the last week or so I have been looking at them in a bit more detail. Vector fields sound quite complicated but they can be very simple. A vector field can be presented visually as a vector attached to each point in space. The space may be the $x$-$y$ plane, three-dimensional space, it could be a region of the $x$-$y$ plane or even a manifold. A typical vector field in 2-dimensions might look as follows. An image of a simple vector field drawn in SAGE Math

The arrows represent the vector that is attached to that particular point – the direction of the arrow gives the direction of the vector and the size of the arrow gives an idea of its relative magnitude. Graphical representations of vector fields can be a little misleading as it is tempting to think that only certain points have vectors attached to them – this is not the case every point has a vector attached to it but if we were to try to show all of them the diagram would be too cluttered.

Vector fields are useful to model flows of liquids or gases for example in weather prediction a vector field that changes over time could be used to model wind patterns. The vector attached to each point would tell you the direction and strength of the wind at that point and the vector field would evolve from one moment to the next. If you define a surface in a vector field then you can use integration to measure the flux across the surface – the physical interpretation of flux is as a measure of the amount of substance flowing across a surface. I came across this question in the book Advanced Engineering Mathematics Fifth Edition by Stroud and Booth and decided to give it a go.

Evaluate $int_mathbf.mathrmmathbf$ over the surface $S$ defined by $x^<2>+y^<2>+z^<2>=4$ for $zgeq0$ and bounded by $x=0$, $y=0$, $z=0$ and $mathbf=xmathbf+2zmathbf+ymathbf$.

The pictures below give an idea of what the vector field and surface both look like from a few different angles. This problem is asking to integrate this vector field over the surface to find the flux across the surface.

For this problem, since the surface that we are integrating over is part of a sphere, it is convenient to change to spherical polar co-ordinates given by$x=rmathrm hetamathrmphi$ $y=rmathrm hetamathrmphi$ $z=rmathrm heta$.The integration itself is quite straightforward although some of the integrands look a bit of a pain at first glance, but some techniques from A-Level Further Maths courses should clear things up. I used some reduction formulae to deal with some of the integrals that I ended up with which really simplified things (which is always good). You can download and view my full solution here – Integrating Vector Fields.

## Calculus of Residues

After blowing off the cobwebs after a couple of years I have been looking at some of the notes that I made some years back on some courses that I took at Warwick on Complex Analysis and Vector Analysis.

Integration has always been one of my favourite areas of mathematics. At A-Level I learned lots of different techniques for calculating some interesting integrals – but A-Level only just skims the surface when it comes to integration and it can be difficult for A-Level students (through no fault of their own) to appreciate the significance of integration. Integration by Parts, Integration by substitution and reduction formula are all great but there are still many integrals which require more advanced techniques to calculate. Contour integrals and the calculus of residues can often come to the rescue.

Contour integrals are a way of passing difficult integrals over a real-interval such as $int_<-infty>^!><1+x^<4>>mathrmx>$ into the complex plane and taking advantage of the Cauchy integral theorem and the calculus of residues. I remember how it felt when I first learned the formula for integration by parts because it meant that I was able to find integrals that were previously impossible for me to calculate – even though I have done contour integrals before it has been very exciting for me to re-discover them. Looking through one of my books I came across this problem – show that for $a>1$

After spending a good deal of one of my afternoons wrestling with the algebra I managed to arrive at a solution which you can download here as a pdf. Here is a graph of the integrand in the case when $a=2$ As you can see from the diagram, the area bounded by the curve and the $x$-axis certainly exists but trying to calculate this integral using A-Level techniques is going to be incredibly difficult if not impossible (if anyone can do it then I would love to see the solution). Unfortunately there are and always will be integrals that cannot be calculated analytically – this is just the way it is and there is no getting around it but contour integrals certainly allows you to calculate a huge range of integrals that previously would have been seemingly impossible.

Contemporary Mathematics
Course Supervisor: John Lund
Basic skills in applicable mathematics including financial matters (simple and compound interest, annuities and loans), trigonometry and some elementary statistics.

College Algebra
Course Supervisor: John Lund Student Success Coordinator: Heidi Staebler-Wiseman
Focus is on using previously learned algebra to model and solve problems, and to explore various types of functions such as linear, quadratic, polynomial, exponential, and logarithmic.

Numbers & Operations for K-8 Teachers
Course Supervisor: Mary Alice Carlson
The study of number and operations for prospective elementary and middle school teachers, including whole numbers, decimals, fractions, percents, integers, operations, numeration systems, and problem solving.

Precalculus
Course Supervisor: Derek Williams
A course designed to produce a deep understanding of algebra and trigonometry so students will be well-prepared for calculus.

Calculus for Technology I
Course Supervisor: John Lund
Calculus with emphasis on problems of interest to engineering technologists. Includes analytic geometry, differentiation, and introduction to integration

Calculus for Technology II
Course Supervisor: John Lund
Calculus with emphasis on problems of interest to engineering technologies. Includes analytic geometry, differentiation, and introduction to integration.

Calculus I Course Supervisor: Jack Dockery
Student Success Coordinator: Veronica Baker
SSC Assistant: Clinton Watton
This first semester of calculus concentrates on the fundamentals of the derivative and its applications: tangent lines to curves optimization problems velocity and acceleration. There is also an introduction to integration with applications to geometry and physics.

Calculus II
Course Supervisor: Jack Dockery
Student Success Coordinator: Veronica Baker
SSC Assistant: Clinton Watton The second semester of calculus covers integration theory, methods of integration, applications of the integral, Taylor's theorem, infinite sequences and series. The course also includes a brief coverage of parametric and polar equations.

Introduction to Statistics
Course Supervisor: Stacey Hancock
Traditional and resistant estimators of location and spread, fundamentals of inference using randomization and classical methods, confidence intervals, and tests of hypotheses

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## Benefits of Math for kids

Besides this, math offers great benefits to the kids. Let us discuss how.

1. As your child pretends to buy and sell toys, he learn math, including counting, arithmetic, problem-solving, and simple money concepts.
2. Math helps make the normal things throughout everyday life, fun, and challenging games like Sudoku and Monopoly, simple.
3. Advanced mathematics encourages the kid to tackle the issue and talk about their findings.
4. Understanding the math concepts will provide children with a strong basis for reasoning, estimation, and thinking logically in later life

Besides helping your child with their everyday mathematics by including it in games, you can also make maths a part of your day-to-day life.

## The lego race

At the end of the 17th century, Gottfried Wilhelm Leibniz (1646-1716) and Isaac Newton (1643-1727), independently one from the other, invented a brilliant mathematical tool: infinitesimal calculus or differential and integral calculus. This is an incredibly efficient crystal ball to predict the future, provided the system in question is governed by a differential equation. This second chapter is about an introduction to the subject in the Lego world. How can we define the speed of a lego man that walks? The average speed is the ratio of the distance travelled, and the time that it took to do so. With that we can calculate the average speed for each step. But what about a driving car? The idea here is to consider the motion of a car as a succession of small steps, so small that they cannot be noticed. This is the basis for the derivative or differential calculus. Imagine a flowing river. For each point of the river, it is possible to calculate the speed of the water at that point. We then take a drawing of the river and draw an arrow on the point in question. The length indicates the speed, and the direction indicates the direction of the speed. Such an arrow is called a vector, and we have such a vector for each point of the river. Mathematicians call this a vector field. Integral calculus is the opposite of differential calculus. The task is now to calculate trajectories in a given vector field. The film shows how lego men moving through a vector field have no choice but to follow a predetermined path. This is known mathematically as the Cauchy-Lipschitz theorem and summarises the concept of determinism: with a given vector field, and a given starting position, there is a unique trajectory starting from that point, and this trajectory is tangent everywhere to the velocity vectors. Determinism as we have defined it has its limits, as we can show with a simple example. In 1879, the physicist James Clerk Maxwell (1831-1879) insisted on the importance of initial conditions for physical phenomena. « There is a maxim…that the same causes will always produce the same effects [. ] There is another maxim which must not be confounded with the first, which asserts that “like causes produce like effects. This is only true when small variations in the initial circumstances produce only small variations in the final state of the system. In a great many physical phenomena this condition is satisfied but there are other cases in which a small initial variation may produce a very great change in the final state of the system. » At the end of this chapter we see our lego men flying in their small spacecrafts. The images should convince you that now, in three dimensions, the situation can become very complicated..

VectorPlot3D expects a function of Cartesian $(x,y,z)$ coordinates, but our functions use spherical coordinates. The solution is use CoordinateTransform . As an example,

We recognize this result as $(r, heta,phi)$ . To make things as simple as possible, let's define our functions like this:

The reason we use those strange definitions is for compatibility with CoordinateTransformation . Now we can pass $(x,y,z)$ arguments to CoordinateTransformation , get back $(r, heta,phi)$ , pass those to ele along with $R$ as a separate argument list and get back a Cartesian vector. Here is one way to plot the vectors: What's going on? First, the @@ tells MMA to replace the head of List[r,θ, ϕ] with ele , so we end up with ele[r, θ, ϕ] . Then we follow that with [R] separately, all of which evaluates to our vector. That's the advantage of those strange definitions -- it lets us use @@ on the first 3 arguments only.

Also, we did not include $sigma$ or $k$ in our functions and didn't use them in our With . That's because VectorPlot3D will rescale the vectors anyway, so it doesn't make any difference. So, really, the With is not necessary. It's only providing a value to $R$ .

## Vector calculus

Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space ( mathbb^3 ). The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow.

Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra, which uses exterior products does generalize, as discussed below.

Basic objects
Scalar fields
Main article: Scalar field

A scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number or a physical quantity. Examples of scalar fields in applications include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

Vector fields
Main article: Vector field

A vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.
Vectors and pseudovectors

In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields except that they change sign under an orientation-reversing map: for example, the curl of a vector field is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction. This distinction is clarified and elaborated in geometric algebra, as described below.
Vector operations
Algebraic operations
Main article: Euclidean vector § Basic properties

The basic algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field, and consist of:

scalar multiplication
multiplication of a scalar field and a vector field, yielding a vector field: ( a old )
addition of two vector fields, yielding a vector field: ( old_1 + old_2 )
dot product
multiplication of two vector fields, yielding a scalar field: (old_1 cdot old_2 )
cross product
multiplication of two vector fields, yielding a vector field: ( old_1 imes old_2 )

There are also two triple products:

scalar triple product
the dot product of a vector and a cross product of two vectors: ( old_1cdotleft( old_2 imesold_3 ight) )
vector triple product
the cross product of a vector and a cross product of two vectors: ( old_1 imesleft( old_2 imesold_3 ight) or left( old_3 imesold_2 ight) imesold_1 )

although these are less often used as basic operations, as they can be expressed in terms of the dot and cross products.

Differential operations
Main articles: Gradient, Curl (mathematics), Divergence and Laplacian

Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator ( abla), also known as "nabla". The five most important differential operations in vector calculus are:

Operation Notation Description Domain/Range
Gradient ( operatorname(f)= abla f ) Measures the rate and direction of change in a scalar field. Maps scalar fields to vector fields.
Curl ( operatorname(mathbf)= abla imesmathbf ) Measures the tendency to rotate about a point in a vector field. Maps vector fields to (pseudo)vector fields.
Divergence ( operatorname
(mathbf)= ablacdotmathbf )
Measures the scalar of a source or sink at a given point in a vector field. Maps vector fields to scalar fields.
Vector Laplacian ( abla^2mathbf= abla( ablacdotmathbf)- abla imes( abla imesmathbf) ) Measures the difference between the value of the vector field with its average on infinitesimal balls. Maps between vector fields.
Laplacian ( Delta f= abla^2 f= ablacdot abla f ) Measures the difference between the value of the scalar field with its average on infinitesimal balls. Maps between scalar fields.

where the curl and divergence differ because the former uses a cross product and the latter a dot product, f denotes a scalar field and F denotes a vector field. A quantity called the Jacobian is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration.
Theorems

Likewise, there are several important theorems related to these operators which generalize the fundamental theorem of calculus to higher dimensions:

Theorem Statement Description
Gradient theorem ( int_ ablavarphicdot dmathbf = varphileft(mathbf ight)-varphileft(mathbf

ight) )

The line integral through a gradient (vector) field equals the difference in its scalar field at the endpoints of the curve L.
Green's theorem ( int!!!!int_ left (frac - frac ight), dmathbf=oint_ left ( L, dx + M, dy ight ) ) The integral of the scalar curl of a vector field over some region in the plane equals the line integral of the vector field over the closed curve bounding the region oriented in the counterclockwise direction.
Stokes' theorem ( int!!!!int_ abla imes mathbf cdot dmathbf = oint_ mathbf cdot d mathbf ) The integral of the curl of a vector field over a surface in ( mathbb R^3 ) equals the line integral of the vector field over the closed curve bounding the surface.
Divergence theorem ( int!!!!int!!!!int_left( ablacdotmathbf ight)dmathbf=oiintscriptstyle partial Vmathbf Fcdotmathbf S ) The integral of the divergence of a vector field over some solid equals the integral of the flux through the closed surface bounding the solid.

Applications
Linear approximations
Main article: Linear approximation

Linear approximations are used to replace complicated functions with linear functions that are almost the same. Given a differentiable function f(x, y) with real values, one can approximate f(x, y) for (x, y) close to (a, b) by the formula

The right-hand side is the equation of the plane tangent to the graph of z=f(x, y) at (a, b).

Optimization
Main article: Mathematical optimization

For a continuously differentiable function of several real variables, a point P (that is a set of values for the input variables, which is viewed as a point in ( mathbb^n )) is critical if all of the partial derivatives of the function are zero at P, or, equivalently, if its gradient is zero. The critical values are the values of the function at the critical points.

If the function is smooth, or, at least twice continuously differentiable, a critical point may be either a local maximum, a local minimum or a saddle point. The different cases may be distinguished by considering the eigenvalues of the Hessian matrix of second derivatives.

By Fermat's theorem, all local maxima and minima of a differentiable function occur at critical points. Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.

Vector calculus is particularly useful in studying:

Center of mass
Field theory
Kinematics

Generalizations
Different 3-manifolds

Vector calculus is initially defined for Euclidean 3-space, ( mathbb^3 ), which has additional structure beyond simply being a 3-dimensional real vector space, namely: an inner product (the dot product), which gives a notion of length (and hence angle), and an orientation, which gives a notion of left-handed and right-handed. These structures give rise to a volume form, and also the cross product, which is used pervasively in vector calculus.

The gradient and divergence require only the inner product, while the curl and the cross product also requires the handedness of the coordinate system to be taken into account (see cross product and handedness for more detail).

Vector calculus can be defined on other 3-dimensional real vector spaces if they have an inner product (or more generally a symmetric nondegenerate form) and an orientation note that this is less data than an isomorphism to Euclidean space, as it does not require a set of coordinates (a frame of reference), which reflects the fact that vector calculus is invariant under rotations (the special orthogonal group SO(3)).

More generally, vector calculus can be defined on any 3-dimensional oriented Riemannian manifold, or more generally pseudo-Riemannian manifold. This structure simply means that the tangent space at each point has an inner product (more generally, a symmetric nondegenerate form) and an orientation, or more globally that there is a symmetric nondegenerate metric tensor and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point.
Other dimensions

Most of the analytic results are easily understood, in a more general form, using the machinery of differential geometry, of which vector calculus forms a subset. Grad and div generalize immediately to other dimensions, as do the gradient theorem, divergence theorem, and Laplacian (yielding harmonic analysis), while curl and cross product do not generalize as directly.

From a general point of view, the various fields in (3-dimensional) vector calculus are uniformly seen as being k-vector fields: scalar fields are 0-vector fields, vector fields are 1-vector fields, pseudovector fields are 2-vector fields, and pseudoscalar fields are 3-vector fields. In higher dimensions there are additional types of fields (scalar/vector/pseudovector/pseudoscalar corresponding to 0/1/n−1/n dimensions, which is exhaustive in dimension 3), so one cannot only work with (pseudo)scalars and (pseudo)vectors.

In any dimension, assuming a nondegenerate form, grad of a scalar function is a vector field, and div of a vector field is a scalar function, but only in dimension 3 and 7  (and, trivially, dimension 0) is the curl of a vector field a vector field, and only in 3 or 7 dimensions can a cross product be defined (generalizations in other dimensionalities either require n-1 vectors to yield 1 vector, or are alternative Lie algebras, which are more general antisymmetric bilinear products). The generalization of grad and div, and how curl may be generalized is elaborated at Curl: Generalizations in brief, the curl of a vector field is a bivector field, which may be interpreted as the special orthogonal Lie algebra of infinitesimal rotations however, this cannot be identified with a vector field because the dimensions differ - there are 3 dimensions of rotations in 3 dimensions, but 6 dimensions of rotations in 4 dimensions (and more generally ( extstyle<inom<2>=frac<1><2>n(n-1)> ) dimensions of rotations in n dimensions).

There are two important alternative generalizations of vector calculus. The first, geometric algebra, uses k-vector fields instead of vector fields (in 3 or fewer dimensions, every k-vector field can be identified with a scalar function or vector field, but this is not true in higher dimensions). This replaces the cross product, which is specific to 3 dimensions, taking in two vector fields and giving as output a vector field, with the exterior product, which exists in all dimensions and takes in two vector fields, giving as output a bivector (2-vector) field. This product yields Clifford algebras as the algebraic structure on vector spaces (with an orientation and nondegenerate form). Geometric algebra is mostly used in generalizations of physics and other applied fields to higher dimensions.

The second generalization uses differential forms (k-covector fields) instead of vector fields or k-vector fields, and is widely used in mathematics, particularly in differential geometry, geometric topology, and harmonic analysis, in particular yielding Hodge theory on oriented pseudo-Riemannian manifolds. From this point of view, grad, curl, and div correspond to the exterior derivative of 0-forms, 1-forms, and 2-forms, respectively, and the key theorems of vector calculus are all special cases of the general form of Stokes' theorem.

From the point of view of both of these generalizations, vector calculus implicitly identifies mathematically distinct objects, which makes the presentation simpler but the underlying mathematical structure and generalizations less clear. From the point of view of geometric algebra, vector calculus implicitly identifies k-vector fields with vector fields or scalar functions: 0-vectors and 3-vectors with scalars, 1-vectors and 2-vectors with vectors. From the point of view of differential forms, vector calculus implicitly identifies k-forms with scalar fields or vector fields: 0-forms and 3-forms with scalar fields, 1-forms and 2-forms with vector fields. Thus for example the curl naturally takes as input a vector field, but naturally has as output a 2-vector field or 2-form (hence pseudovector field), which is then interpreted as a vector field, rather than directly taking a vector field to a vector field this is reflected in the curl of a vector field in higher dimensions not having as output a vector field.

Real-valued function
Function of a real variable
Real multivariable function
Vector calculus identities
Del in cylindrical and spherical coordinates
Directional derivative
Irrotational vector field
Solenoidal vector field
Laplacian vector field
Helmholtz decomposition
Orthogonal coordinates
Skew coordinates
Curvilinear coordinates
Tensor

There is also the perp dot product, which is essentially the dot product of two vectors, one vector rotated by π/2 rads, equivalently the magnitude of the cross product:

( old_1 ot cdotold_2 = left | old_1 imes old_2 ight | = left | old_1 ight | left | old_2 ight | sin heta, )

where θ is the included angle between ( v_1 ) and ( v_2 ). It is rarely used, since the dot and cross product both incorporate it.

Galbis, Antonio & Maestre, Manuel (2012). Vector Analysis Versus Vector Calculus. Springer. p. 12. ISBN 978-1-4614-2199-3.

Weisstein, Eric W. "Perp Dot Product." From MathWorld--A Wolfram Web Resource.

Sandro Capparini (2002) "The discovery of the vector representation of moments and angular velocity", Archive for History of Exact Sciences 56:151–81.
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J.E. Marsden (1976). Vector Calculus. W. H. Freeman & Company. ISBN 0-7167-0462-5.
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Barry Spain (1965) Vector Analysis, 2nd edition, link from Internet Archive.
Chen-To Tai (1995). A historical study of vector analysis. Technical Report RL 915, Radiation Laboratory, University of Michigan.

Hazewinkel, Michiel, ed. (2001), "Vector analysis", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Hazewinkel, Michiel, ed. (2001), "Vector algebra", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Vector Calculus Video Lectures from University of New South Wales on Academic Earth
A survey of the improper use of ∇ in vector analysis (1994) Tai, Chen
Expanding vector analysis to an oblique coordinate system
Vector Analysis: A Text-book for the Use of Students of Mathematics and Physics, (based upon the lectures of Willard Gibbs) by Edwin Bidwell Wilson, published 1902.
Earliest Known Uses of Some of the Words of Mathematics: Vector Analysis