2012年8月29日 · Finding the basis vectors We can index all of the points on the manifold with arbitrary coordinates x [itex]\mu[/itex]. A curve on a manifold is a mapping from the real numbers to that manifold - we can define such a curve C as x [itex]\mu[/itex] = x [itex]\mu[/itex] (t), where t is a real number and [itex]\mu[/itex] ranges over all the ...

2006年10月5日 · In modern notation, basis vectors are usually written e i, where i = 0...3. I.e. basis vectors a boldfaced, and subscripted. Hatted vectors usually imply a non-coordinate basis. Hatted vectors are usually derived from an orthonormal basis of one-forms, also sometimes called a coframe. They have a very "physical" interpretation.

2024年11月5日 · The figures show that basis vectors and dual basis vectors with different indices are perpendicular to each other, so their dot product equals zero. So far, so good. Then, in order for the dot product between basis vectors and dual basis vectors with the same index to equal one, they should be in the same direction.

2021年12月13日 · (A scalar times a vector is a vector. And I assume basis vectors are vectors.) Hence, what we are doing to get the vector is not adding the vector components but the vector components times the basis vectors and then add the product of the vector component times the basis vector vectorially to get a vector. Is this how it's done?

2014年8月22日 · These basis vectors are used to describe vectors in a polar coordinate system, which is a different way of representing points in space. In order to describe a vector using basis vectors, you need to understand that the basis vectors are defined at every point in space. This means that at each point, there are two basis vectors - one for the ...

2018年4月14日 · Basis vectors are used to calculate the inner product of two vectors by providing a coordinate system for each vector. What is the significance of basis vectors and inner product in mathematics? Basis vectors and inner product are fundamental concepts in linear algebra and are used in a wide range of fields such as physics, engineering, and ...

2010年4月26日 · Hi, I'm having trouble understanding how people can make calculations using the partial derivatives as basis vectors on a manifold. Are you allowed to specify a scalar field on which they can operate? eg. in GR, can you define f(x,y,z,t) = x …

2013年5月6日 · A lattice vector is [itex]\vec{R} = n_1\vec{a}_1 + n_2\vec{a}_2 + n_3\vec{a}_3[/itex] where a1, a2 and a3 are the basis vectors (n's are integers). Generally there are three basis vectors, these form a linearly independent set from which you can construct any lattice vectors. The set of lattice vectors is the set of all lattice points in space.

2015年10月2日 · Thanks, I thought that in the explanation I quoted (p. 62, 1st edition) he was going to into more detail about how this new definition comes about, e.g. it has its origins in the tensor that defines the metric which introduces the minus sign, and that the components of this tensor are defined by the basis vectors of the space.

2019年8月22日 · It is related to the basis vectors through the inner product operation. The metric tensor can be used to calculate the length and angle between basis vectors, providing a way to measure distances and angles in a curved spacetime. In this way, the metric tensor and basis vectors are closely connected in general relativity.