# Covariant and contravariant vector

1. Oct 8, 2012

### Shan K

Will any one help me to under stand the covariant and contravariant vector ? And can any one show me the derivation of
dr=(dr/dx)dx + (dr/dy)dy + (dr/dz)dz

2. Oct 8, 2012

### atyy

Last edited by a moderator: Sep 25, 2014
3. Oct 8, 2012

### HallsofIvy

Staff Emeritus
The last, at least, is just the "chain rule" from Calculus. Suppose r is a function of x, y, and z and they, themselves, depend on the parameter t. We can think of r as, perhaps, the radius of curvature of a surface defined by x, y, and z, and (x(t), y(t), z(t)) as the trajectory of an object moving over that surface. Of course, then we can think of r(t)= r(x(t), y(t), z(t)) as giving the
radius of curvature at each point along that trajectory.

By the chain rule,
$$\frac{dr}{dt}= \frac{\partial r}{\partial x}\frac{dx}{dt}+ \frac{\partial r}{\partial y}\frac{dyz}{dt}+ \frac{\partial r}{\partial z}\frac{dz}{dt}$$

In terms or "differentials" we can write
$$dr= \left(\frac{\partial r}{\partial x}\frac{dx}{dt}+ \frac{\partial r}{\partial y}\frac{dy}{dt}+ \frac{\partial r}{\partial z}\frac{dz}{dt}\right)dt= \frac{\partial r}{\partial x}\frac{dx}{dt}dt+ \frac{\partial r}{\partial y}\frac{dy}{dt}dt+ \frac{\partial r}{\partial z}\frac{dz}{dt}dt$$
$$dr= \frac{\partial r}{\partial x}dx+ \frac{\partial r}{\partial y}dy+ \frac{\partial r}{\partial z}dz$$

Last edited: Oct 8, 2012
4. Oct 8, 2012

### cosmik debris

Do a search, this comes up quite frequently. There are some excellent explanations in the Linear Algebra forum.

5. Oct 8, 2012

### dydxforsn

Yeah, as Cosmik Debris has stated, this topic really does come up a lot. I would recommend also searching the "Special and General Relativity" subforum as well as "Topology/Differential Geometry". Really you are looking for any area where generalized curvilinear coordinate representations (as opposed to just straight cartesian coordinates - 'x', 'y', and 'z') are regularly used. That's kind of the field when this stuff gets taught, the study of generalized curvilinear coordinate systems (this subject also might be worth a wikipedia search for the OP.)

Also, recently I remember making this post to explain the meaning of a dual vector (also known as a covariant vector or a 1-form. 1-forms and "vectors" are the modern terminology taught in differential geometry for covariant and contravariant vectors respectively. Not to confuse you, but both satisfy the properties for a vector space as taught in abstract algebra, but in the modern terminology only one gets the distinction of being associated with the terminology, "vector" heh): https://www.physicsforums.com/showthread.php?p=4057781#post4057781

I think that's about as simple an explaination on the differences (and necessities) of contravariant and covariant vectors as you'll find.

And as Halls said the other thing you mentioned is just a chain rule from calculus that looks slightly weird but does come up quite a bit.