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 somone 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.

 

[asteriatic]

Sure, I'd be happy to explain the difference between displacement and gradient vectors.

Displacement vectors represent the change in position of a point or object in space. They are typically represented by an arrow pointing from the initial position to the final position of the object. These vectors are contravariant, meaning they change direction when the coordinate system is rotated.

On the other hand, gradient vectors represent the change in a scalar quantity, such as temperature or pressure, in a given direction. They are typically represented by a vector with magnitude and direction, pointing in the direction of maximum increase of the scalar quantity. These vectors are covariant, meaning they do not change direction when the coordinate system is rotated.

In terms of derivatives, the gradient vector is the covariant form of the derivative, while the displacement vector is the contravariant form. This means that when calculating derivatives in different coordinate systems, the gradient vector will remain the same, while the displacement vector will change direction.

I hope this helps clarify the difference between these two types of vectors. Let me know if you have any further questions.