You can't have the same index three times! Also, using the "summation convention" you sum over one "upper" and one "lower" index so even if you wrote [itex]g^{kj}B^{ij}_k[/itex], you would not be summing. Perhaps what you mean is [itex]g^k_jB^{ij}_k= B^i[/itex]cristina89 said:Homework Statement
B is a third order tensor. Show that [itex]B^{ij}_{i}[/itex] is a contravariant vector.
The Attempt at a Solution
Well... I just thought about a simple solution but I don't think I'm right. But anyways.
Considering [itex]B^{ij}_{i}[/itex]. If I raise the index i: [itex]g^{ij}B^{ij}_{i} = B^{ijj}[/itex]
Well, what is the definition of a contravariant vector? Can you show that this satisfies that definition? Remember that it is NOT enough just to show that something can be written with a single index- you can write an array, in a given coordinate system, indexed with a single index- that does not make it a "vector". A vector must change coordinates correctly as the coordinate system changes. If I remember correctly (it's been a while!) one test for a contravariant vector, written as [itex]v^i[/itex], is that the combination [itex]g_{ij}v^iv^j[/itex] (essentially the vector length) is a scalar. Which, again, does not just mean "a number" in a given coordinate system. A scalar (0 order tensor) must not change when you change coordinate systems.And so I can say that B is a first order tensor = vector. And this is a contravariant vector.
Is this right?
A contravariant vector is a type of mathematical object used in vector calculus and tensor analysis. It represents a quantity that changes in a specific direction when the coordinate system is transformed.
While both contravariant and covariant vectors are used to represent quantities that change with respect to coordinate transformations, they differ in how they transform under those transformations. A contravariant vector changes in the opposite direction of the coordinate system, while a covariant vector changes in the same direction.
In physics, contravariant vectors are used to represent physical quantities such as velocity, acceleration, and force. They are essential in the study of motion and dynamics, as they allow for the analysis of how these quantities change in different coordinate systems.
Contravariant vectors are closely related to tensors, as they are one of the components that make up a tensor. A tensor is a mathematical object that can represent multiple quantities and their relationships in different coordinate systems.
Contravariant vectors can be visualized as arrows or directed line segments in a specific direction. However, unlike covariant vectors which can be visualized as tangent vectors on a curved surface, contravariant vectors are not as easily visualized in physical space.