Covariant and contravariant vecotr questions

[drlang]

Hi

I am trying to learn about covariant and contravariant vectors and derivatives. The videos I have been watching talk about displacement vector as the basis for contravariant vectors and gradient as the basis for covariant vectors. Can someone tlel me the difference between displacemement and gradient vectors?

 

[Fredrik]

This post and the ones linked to at the end explain some of what you need to know.

The term "displacement vector" doesn't make much sense to me. They are tangent vectors, nothing more, nothing less.

 

[drlang]

Hi Fredrik

I am using the terms he used.

Can you clarify the difference between tangent vector and gradient vector. Maybe that would help me.

 

[Fredrik]

I don't think "gradient vector" is a standard term, and I'm not sure what they mean by it. The gradient of a real-valued function f is defined as the tangent vector field with components $\frac{\partial f}{\partial x^i}$, if we're talking about functions defined on $\mathbb R^n$. So maybe that's what they mean. If we're talking about some arbitrary smooth manifold with a metric, then the gradient is defined as the vector field with components $g^{ij}\frac{\partial f}{\partial x^j}$, where the $g^{ij}$ are the components of the inverse of the matrix of components of the metric, and the partial derivatives with respect to a coordinate system x are defined in one of the posts I linked to.

If you're only concerned with $\mathbb R^n$, and not arbitrary smooth manifolds with metrics, then some things get simpler, but I still think it's worth the time read the posts I linked to, because they will make it easier to understand the terminology, and what's really going on in some definitions and calculations.