Vector Calculus 2
Nisha Heidinger
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, R 3 . \displaystyle \mathbb R ^3. The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans 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 the theory of quaternions 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 its standard form using the cross product, vector calculus does not generalize to higher dimensions, but the alternative approach of geometric algebra, which uses the exterior product, does (see Generalizations below for more).
A scalar field associates a scalar value to every point in a space. The scalar is a mathematical number representing 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 (known as scalar bosons), such as the Higgs field. These fields are the subject of scalar field theory.
A vector field is an assignment of a vector to each point in a space.[1] 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. This can be used, for example, to calculate work done over a line.
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.
The algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then applied pointwise to a vector field. The basic algebraic operations consist of:
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 initially defined for Euclidean 3-space, R 3 , \displaystyle \mathbb R ^3, which has additional structure beyond simply being a 3-dimensional real vector space, namely: a norm (giving a notion of length) defined via an inner product (the dot product), which in turn gives a notion of 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 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; 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.
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.
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 or 1-form, 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.
This is a text on elementary multivariable calculus, designed for studentswho have completed courses in single-variable calculus. The traditional topicsare covered: basic vector algebra; lines, planes and surfaces; vector-valuedfunctions; functions of 2 or 3 variables; partial derivatives; optimization;multiple integrals; line and surface integrals.
The book also includes discussion of numerical methods: Newton's method foroptimization, and the Monte Carlo method for evaluating multiple integrals.There is a section dealing with applications to probability. Appendices includea proof of the right-hand rule for the cross product, and a short tutorial onusing Gnuplot for graphing functions of 2 variables.
(2022-08-15) Corrected a few references to equation numbers in Section 4.3, and replaced a theta by phi in the spherical coordinateversion of the divergence in Section 4.4 (thanks to Yanfeng Li of Tianjin University).
When making the corrections I realized that I will need to update the LaTeX code so that it compiles with TeXLive 2020 (the compilation fails miserably right now). I was able to compile the PDF using TeXLive 2011 but it required some hacks. This might be a good time to redesign the book in general.
(2021-01-05) Cleaned up the web pageto make it less hideous and more consistent with the revamped page forElementary Calculus.(2021-01-02) After too many years of neglectI finally got around to correcting a few typos that had been floating out there for a while:
- Appendix A: The answer to Exercise 5 from Section 1.9 is now fixed.
- Section 1.1: In Example 1.3(d) R^3 now has the correct dimension.
- Section 1.7: In Example 1.33 a minus is now a plus.
- Appendix A: The margin overrun in the answers for Section 2.4 has been removed.
Problems with my TeXLive 2014 setup had caused numerous issues when trying to compilethe book. I ended up going back to TeXLive 2011 to fix all that, and it worked. So now, aftermany requests, I have finally restored the ability to buy a printed and bound paperback version on Lulu.com. It's even a buck cheaper than before. See the link near the top of this page.On a side note, there are many things about the book that I would change now, after theexperience of writing the Trigonometry book and especially the Elementary Calculusbook, both content-wise and stylistically. I haven't decided on that yet, but if I do re-writeVector Calculus then I would keep the current version available in addition to the newversion. Any decision on that wouldn't be for at least another year, though.
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