Our vector space may come equipped with a metric that is derived from a nondegenerate inner product. We wish to write the velocity vector at some time t 0. Covariance and contravariance of vectors 3 for some invertible n. But well do a bit more on this later take another vector. Packed with examples, including matlab examples, this book will show you. In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. Within the general scheme for the construction of tensors, a covariant vector is identified with a covariant tensor of valency 1. Theres no reason why we cant have a oneform that satisfies this definition. But the resolution is simply that and correspond to different objects, a dual vector and vector respectively. A contravariant vector or tangent vector often abbreviated simply as vector, such as a direction vector or velocity vector has components that contravary with a change of basis to compensate.
Tensors, contravariant and covariant components of a vector. Tk mn rp is a mixed tensor with contravariant rank 4 and covariant rank 2. But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system. That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. Yes, and sometimes youll hear the two ways of writing the single vector as contravariant representation and covariant representation. What is the geometric interpretation of the contravariant and covariant vectors if we define them using the equation that i pasted. This fact that a covariant vector like w can convert any contravariant vector like v into a scale invariant value like f is summarized by saying that w is a linear functional. Relationship between covariantcontravariant basis vectors.
How to prove any vector identity using cartesian tensors. Mathematical operations, such as converting one sort of vector into another sort of vector, are done on vector spaces. Consider the expression j i a i x where j is free index. Vectors, metric and the connection 1 contravariant and covariant vectors 1. Feb 17, 2010 vector is a physical quantity and it does not depend on any coordinate system. One contravariant, v i is based on the notion of building v by adding together the basis vectors ei.
Tensors covariant differential and riemann tensor coursera. In the ordered pair, an element of is called a contravariant vector. Vector is a physical quantity and it does not depend on any coordinate system. As mentioned in a previous video, tensors are invariant under coordinate transformations.
Oct 20, 2016 covariant differentiation for a contravariant vector. For the contravariant components it is customary to use superscripts for the indices such as j and k. Covariant differentiation for a contravariant vector. Using the product rule of derivation, the rate of change of the components v. Abstract vector is a physical quantity and it does not depend on any coordinate system.
Remember that summation is implied over the repeated index u, whereas the index v appears only once in any given product so this expression applies for any value of v. Vectors, metric and the connection 1 contravariant and. Mar 23, 2019 in this video, i describe the meaning of contravariant and covariant vector components. Covariant and contravariant bases are dual to one another and are physics nomenclature for constructs that arise in differential geometry. Covariant and contravariant vectors can be interconverted with use of the metric tensor, given by9.
How to use matlab to calculate dot and cross products, and solve linear equations. Now, the only question i have left is, what is gained by talking about contravariance and covariance, rather than just vector spaces and their dual spaces. Distinction between covariant and contravariant vectors. Change of basis, covariant vectors, and contravariant vector. If i have covariant, but multiplying by this, i obtain contravariant vector. An element of the vector space dual to an dimensional vector space, that is, a linear functional linear form on. The most important point is that both vi and v i are representations of the one vector v. Thus standard vectors are contravariant with respect to changes in the basis. The vector space or linear space, mve4 space, or just space of all kcontravariant, covariant tensors tensors of valence k at the point p in a manifold m will be denoted tk mp, with tmp and t.
Im reading a quora answer on an intuitive explanation of covariantcontravariant components of vectors. And does covariance and contravariance of vectors have any relation to the contravariant functor mapping a vector space v to its dual v. The vector space or linear space, mve4 space, or just space of all k contravariant, covariant tensors tensors of valence k at the point p in a manifold m will be denoted tk mp, with tmp and t. In this video, i describe the meaning of contravariant and covariant vector components. Yes, confusingly, both a vector and its dual would usually be written as a in index free notation. Contravariant vector an overview sciencedirect topics. The second is a brief introduction to coordinate free geometry, which necessitates a discussion of contravariant and covariant vectors. Suppose that the form corresponds in this manner to the tensor and that the form corresponds to its covariant. The components of a covariant vector transform like a gradient and obey the transformation law. So whether a vector is treated as contravariant or covariant is a matter of choice.
A case that was particularly often considered is when is a form in one single contravariant vector. Now lets consider a vector x whose contravariant components relative to the x axes of figure 2 are x 1, x 2, and lets multiply this by the covariant metric tensor as follows. That is, the coordinate axes reverse direction when the conventional righthanded coordinate system becomes a lefthanded coordinate system, where. It seemed covariant and contravariant transformations were very different concepts. Useful expressions for the contravariant coordinates of ov are, using the sine rule, 4 covariant coordinates the covariant components of a vector are defined by the transformation 5. The g term accounts for the change in the coordinates. This is the second volume of a twovolume work on vectors and tensors. General relativitycontravariant and covariant indices. Scalars, vectors and tensors a scalar is a physical quantity that it represented by a dimensional number at a particular point in space and time. A polar vector, also termed a contravariant vector or vector, such as the position vector r, reverses sign when the coordinate axes are reversed. Nonindexed lower or upper case bold face latin letters e.
If you have been confused by vectors, vector calculus, tensor analysis, or quaternions, this book is for you. If it were a tensor, it would be zero in all frames. These results lead us to adopt the following definition. We regard the inner product as being a bilinear form g. Contravariant vector article about contravariant vector by. We may also use it as opposite to scalar and vector i. A gentle introduction to tensors washington university in. If three quantities of a coordinate system are related to three other quantities of another coordinate system by the transformation equations 1 or 2, then the quantities are called components of a contravariant vector or a contravariant tensor of the first rank. Covariant and contravariant vectors alok kumar1 iiser, bhopal iti campus gas rahat building govindpura, bhopal 23 india. So, so if we have two tensors, metric tensor and inverse metric tensor, to every contravariant vector, with the use of the metric tensor we can define corresponding covariant vector. The covariant derivative of the r component in the q direction is the regular derivative plus another term. The duality between covariance and contravariance intervenes whenever a vector or tensor quantity is represented by its components, although modern differential geometry uses more sophisticated index free methods to represent tensors. These two topics will be then combined to write down the formalism for special relativity. The expansion in orthonormal basis is mathematically simple.
Jul 31, 2015 the gradient component of the equation is the contravariant vector, and the partial derivative component of the equation is the covariant vector. In almost all cases, the meaning should be obvious from the context. Vector algebra is an essential physics tool for describing vector quantities in a compact fashion. Old math with a new notation we think of a vector in ordinary threedimensional space as an object with direction and. Chapter 10 vectors and tensors georgia institute of. In curved space, the covariant derivative is the coordinate derivative of the vector, plus the change in the vector caused by the changes in the basis vectors. Verify the following claim in the proof of proposition 4. What is contravariant and covariant vector, tensor in. Here, each vector y j of the f basis is a linear combination of the vectors x i of the f basis, so that contravariant transformation a vector v in v is expressed uniquely as a linear combination of the elements of the f basis as 2. The length of a physical vector must have physical units. A vector called a contravariant vector represents a point in the euclidean space, or primary space, from the origin to the point. The idea of a covariant derivative of a vector field a in the direction a.
Nov 24, 20 the mystery of light walter lewin july 19, 2005 duration. The gradient component of the equation is the contravariant vector, and the partial derivative component of the equation is the covariant vector. The problem here is that physicists often need to use differential geometry for example, for relativity long before they have seen a proper course on differential geometry. It need to be expanded in some basis for practical calculation. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The covariant derivative of the r component in the r direction is the regular derivative. A vector is a bookkeeping tool to keep track of two pieces of information typically magnitude and direction for a physical quantity. In the classical theory of invariants was called the covariant of. If we have a coordinate system with straight coordinate axes, the geometric explanation given is that a vectors covariant components in such a system will be perpendicular projections on the axes, whereas its contravariant components will be parallel projections. Contravariant and covariant vectors physics forums.
This volume begins with a discussion of euclidean manifolds. The more complicated case is for a dual vector or functional, things work in the opposite way because a dual vector is a function. Volume 1 is concerned with the algebra of vectors and tensors, while this volume is concerned with the geometrical aspects of vectors and tensors. Request pdf covariant and contravariant vectors vector is a physical quantity and it does not depend on any coordinate system.
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