The second is a brief introduction to coordinatefree geometry, which neces sitates a discussion of contravariant and covariant vectors. This property is called that transformation of contravariant vectors is transitive. Also note that we take as transformation matrix the matrix that. What we have been using thus far in these lecture notes are contravariant 4vectors a, denoted by the superscript aaaaa 01 2 3,, contravariant 4 vector. Feb 17, 2010 vector is a physical quantity and it does not depend on any coordinate system. You can verify that the normal vector has covariant components by recalling that the normal can be defined through a cross product of tangent vectors which have contravariant components. Covariant versus contravariant vectors complexvariable. A coordinate transformation is linear map from a vector to itself with a change of basis. This important requirement is what distinguishes a contravariant vector from any other triple of physically meaningful quantities.
The components of a vector will always transform with the opposite transformation to the one that transformed the basis, regardless of what specific basis is that. But in many physical situations we have to choose an nonorthogonal basis or oblique coordinate system. There seems to be a real difference in the nature of these vectors, as opposed to simply the components and bases used to express them. When v is written with contravariant components, eqn. The basis for these names will be explained in the next section, but at this stage it is just a name used to distinguish two types of vector. We now examine the dual or covariant vector counterpart to the contravariant vector. The vector concepts and techniques described in the previous chapters are important for two reasons. At q, over new england, its velocity has a large component to the south. The \stick is imbedded in the \lasagna and the inner product is equal to the. We can check that we have correctly computed the hodge star of f by taking. Here, we define contravariant and covariant vector components according to their trans. Theres no reason why we cant have a oneform that satisfies this definition. A vector space is a set whose elements satisfy certain axioms.
May 05, 20 yet, elsewhere i read that the gradient is a covariant vector, and that velocity is a contravariant vector, because of the way the components transform. We wish to write the velocity vector at some time t 0. An introduction to vectors and tensors from a computational. If the contravariant vector a increases incrementally in length from its endpoint p by differential change d a, then the components of d a are d a 1 and d a 2 are also projections to the skewed axes x 1 and x 2, which are obtained by dropping lines perpendicular to the axes from the tip of the incremental vector d a onto the coordinate axes. What different between covariant metric tensor and.
The vector components in a given basis are equivalent to the vector itself, since it is a simple matter to calculate the invariant magnitude and direction from known values of a a a 1 2 3. An example of contravariant tensors is the displacement vector. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. Affine covariantcontravariant vector forms for the. One is called the contravariant vector or just the vector, and the other one is called the covariant vector or dual vector or. Consider the category of pointed topological spaces, i. Pdf covariance and contravariance of vectors semantic. On the other hand, the covariant derivative of the contravariant vector is a mixed secondorder tensor and it transforms according to the transformation law. As mentioned in a previous video, tensors are invariant. Contravariant transforms for a contravariant vector to be coordinate system invariant, the components of the vector must transform oppositely contra as in against, like contrary to the change of the basis vectors the axes to cancel the change. What is contravariant and covariant vector, tensor in. The vector is then expressed as a linear combination of the base vectors. How it transforms depends on the basis you chose for that space, and not on the nature of the vector.
This basis, as discussed at the beginning of this article, is also called the covariant curvilinear basis. A covariant 4 vector a is denoted by its subscript aaaaa 01 2 3,, covariant 4 vector. Integral contravariant form of the navierstokes equations. A vector v red represented by tangent basis vectors yellow, left. Myers components are identi ed with superscripts like v, and covariant vector components are identi ed. Unlike the infinitesimal calculus, tensor calculus allows. The stress at a point of an elastic solid is an example of a tensor which depends on two directions one normal to the area and other that of the force on it.
Many physical quantities are naturally described in terms of vectors, e. Tensors, contravariant and covariant components of a vector. Contravariant vector transform the contravariant vector is a mathematical object whose representation in terms of components transforms according to this rule. One is called the contravariant vector or just the vector, and the other one is called the covariant vector or dual vector or one vector. Threedimensional, navierstokes equations, contravariant, integral form, timedependent curvilinear coordinate system, shockcapturing scheme. 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. Then any vector a can be expressed with respect to this coordinate system in terms of its base vectors as where c 1, c 2, c 3 are called the contravariant components of a. Affine covariantcontravariant vector forms for the elastic. Covariant and contravariant vector components chapter 4. Technically contravariant vectors are in one vector space, and covariant vectors are in a different space, the dual space. In mathematics, tensor calculus, tensor analysis, or ricci calculus is an extension of vector calculus to tensor fields tensors that may vary over a manifold, e. Examples of how you can picture contravariant and covariant vectors. The velocity vector in equation 3 corresponds to neither the covariant nor contravariant form, but is in its socalled physical form that we would measure.
The expansion in orthonormal basis is mathematically simple. A contravariant vector field v on m associates with each chart x a collection of n smooth realvalued coordinate functions v i of the n variables x 1, x 2. Tensors, contravariant and covariant components of a. Moreover, one can easily check that the dual of a dual vector is the original vector, i. Contents i covariant and contravariant components of a vector.
Pdf a guide on tensors is proposed for undergraduate students in physics or engineering. On the other hand, for instance, a triple consisting of the length, width, and height of a rectangular box could make up the three components of an abstract vector, but this vector would not be contravariant, since a change in coordinates on the space does not. You can check this for yourself by writing out the lorentz transformation in matrix form. If c i is covariant and v j is contravariant, then c k v k is a scalar. But certain quantities are associated with two or more directions, such a quantity is called tensor. Example of vector which is not contravariant is arguably not a vector currently. Vectors, metric and the connection 1 contravariant and. We can also define a reciprocal basis, or contravariant curvilinear basis, b i. Contravariant vector article about contravariant vector. The map which assigns to every vector space its dual space and to every linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself. Now there are physical entities that satisfy these properties, which may not be arrows. Curvilinear coordinates mechanical engineering university of utah. The conventional notation represents only the object, ak, without indicating the coordinate system. Covariance and contravariance computer science wikipedia.
To clarify this discussion of transformations, the coordinate system. First we will explain the distinction between the covariant and contravariant components of vectors, thinking of vector fields where a vector is defined at a point rather than as a. When studying tensor calculus the distinction between covariance and contravariance may be obscure and is rarely explained visually. The alternating tensor can be used to write down the vector equation z x.
The components of a contravariant vector transform like a coordinate di. In this video, i define contravariant and covariant vector components. Aug 23, 2020 at p, the planes velocity vector points directly west. The vector field b i is tangent to the q i coordinate curve and forms a natural basis at each point on the curve. A given velocity vector for example has whatever direction and magnitude it has, independent of the coordinate system we use to express it. Pictorial representation of the inner product between a contravariant vector and a covariant 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. While the position vector r and the gradient vector. Covariant versus contravariant vectors physics forums. Neither ilist nor ilist is a subtype of the other, because ilist is invariant on t. As mentioned in a previous video, tensors are invariant under coordin. Chapter 3 cartesian tensors department of applied mathematics. Its not the vector itself that is contravariant or covariant, its the set of components that you form through its parallel or perpendicular projections. This is not the best of terminology, since it gives the impression that the vector is intrinsically covariant or contravariant, when it is in fact only a matter of which base vectors are being used to describe the vector.
To get the velocity we look at the position at a slightly. Contravariant vector an overview sciencedirect topics. For the contravariant components it is customary to use superscripts for the indices such as j and k. The components of a contravariant vector transform like a. Vector is a physical quantity and it does not depend on any coordinate system. In this video, i describe the meaning of contravariant and covariant vector components. Is it foolish to distinguish between covariant and. People commonly talk about contravariant and covariant vectors and tensors, when they really mean contravariant and covariant components. The contravariant vector is a \stick whose magnitude is the length of the stick, while the covector is like a serving of lasagna. Within the general scheme for the construction of tensors, a covariant vector is identified with a. This question points out a problem with the terminology. Pdf covariance and contravariance of vectors semantic scholar. Lets check that this is a valid tensor operation, and see how it transforms. Be sure to verify that your contravariant components agree.
Whenever a vector should be invariant under a change of basis, that is to say it should represent the same geometrical or physical object having the same magnitude and direction as before, its components must transform. Examples of how to use contravariant in a sentence from the cambridge dictionary labs. Covariant and contravariant tensors physics forums. 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. 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. The components of a covariant vector transform like a gradient and obey the transformation law. Further, the domain of the v i is the whole of the range of x.
Check whether the accelerated method gives the same answer. These and other pictorial examples of visualizing contravariant and covariant vectors are discussed in am. In the pages that follow, we shall see that a tensor may be designated as contravariant, covariant, or mixed, and that the velocity expressed in equation 2 is in its contravariant form. The magnitude of the covector is the density of lasagna noodles. Jul 31, 2012 the a basis vectors scale in the same way the coordinates do whereas the contravariant component scale inversely. Since the path is a geodesic and the plane has constant speed, the velocity vector is simply being paralleltransported. The inverse of a covariant transformation is a contravariant transformation. For example, if v consists of the x, y, and zcomponents of velocity, then v is a contravariant vector. Verify the following claim in the proof of proposition 4. The subtyping is reversed because action is contravariant on t. A vector is a geometric quantity that has a magnitude and direction. Covariance and contravariance of vectors wikipedia. But there is a clear 11 correspondence between the space and its dual, and we tend to think of the contravariant and covariant vectors as different descriptions of the same vector.
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