A Euclidean vector sometimes called a geometric or spatial vector, or—as here—simply a vector is a geometric object that has magnitude or length and direction and can be added to other vectors according to vector algebra. Vectors play an important role in physics: velocity and acceleration of a moving object and forces acting on it are all described by vectors. Many other physical quantities can be usefully thought of as vectors.
The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors. In the Cartesian coordinate system, a vector can be represented by identifying the coordinates of its initial and terminal point.
Typically in Cartesian coordinates, one considers primarily bound vectors. The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion. Formulate properties of the dot product, including the algebraic and geometric methods used to calculate it. The dot product takes two vectors and returns a single value. The dot product can only be taken from two vectors of the same dimension. The dot product is the sum of the product of the corresponding parameters. Geometrically, the dot product is the product of the magnitudes of two vectors and the cosine of the angle between them.
This is different from the cross product, which gives an answer in vector form. Dot Product : When finding the dot product geometrically, you need the magnitudes of the vectors and the angle between them.
The cross product of two vectors is a vector which is perpendicular to both of the original vectors. The cross product is a binary operation of two three-dimensional vectors. The result is a vector which is perpendicular to both of the original vectors.
Because it is perpendicular to both original vectors, the resulting vector is normal to the plane of the original vectors. The cross product is different from the dot product because the answer is in vector form in the same number of dimensions as the original two vectors, where the dot product is given in the form of a single quantity in one dimension.
The geometric method of finding the cross product uses the magnitudes of the vectors and the sine of the angle between them:. The algebraic method of finding the cross product of two vectors involves inputting the vector information into matrices and manipulating them:. A line is a vector which connects two points on a plane and the direction and magnitude of a line determine the plane on which it lies. A line is described by a point on the line and its angle of inclination, or slope.
Every line lies in a plane which is determined by both the direction and slope of the line. A line is essentially a representation of a cross section of a plane, or a two dimensional version of a plane which is a three dimensional object. The surface is formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder.
The surface area and the volume of a cylinder have been known since antiquity. In common use, a cylinder is taken to mean a finite section of a right circular cylinder, i. Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
Multivariable Calculus and Vector Analysis
While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries. A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number, depending on which side of the reference plane faces the point.
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.
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Mean value theorem Rolle's theorem. Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem. Fractional Malliavin Stochastic Variations. Glossary of calculus. Main article: Scalar field. Main article: Vector field. Main article: Vector calculus identities. Main articles: Gradient , Divergence , Curl mathematics , and Laplacian. Main article: Linear approximation.
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Please help improve this section by adding citations to reliable sources. If you really think they'll work for everybody, I'm suspicious. Nice to see Keisler's "An infinitesimal approach to calculus" in the list. Great post. Nice one micromass! Well, I have a great reference now:woot:. You must be logged in to post a comment. We often get questions here from people self-studying mathematics.
One of those questions is what of mathematics should I study and in what order. So in order to answer those questions I have decided to make a list of topics a mathematician should ideally know and what prerequisities the topics have.. This book takes you from elementary calculus to the standard topics in multivariable calculus. It even does this in two approaches, namely the standard and the nonstandard approach.
The nonstandard approach came first historically and involves infinitesimal numbers. The tools of infinitesimals were used by many great mathematicians such as Euler and Gauss. Lately, they have fallen into disuse because mathematicians only want to work with real numbers the standard approach. Nevertheless, the mathematician Robinson has shown infinitesimals to be completely rigorous, and they are actually still used in physics and engineering and it provides intuition in pure mathematics. So it is very beneficial to learn the nonstandard approach.
While everybody thinks this book is valuable, many find it dangerous to teach the nonstandard approach only. This criticism is unfounded because the book treats both approaches. Many concepts like logarithms and trigonometric functions are even revised along the way. That said, a familiarity with proofs is recommended. This does entire calculus from a rigorous point of view. The theory is constructed the right way, and the exercises are very interesting.
Especially the historical topics are very interesting. After seeing single-variable calculus more rigorously, you might want to like to see multivariable calculus more rigorously.