Can I get a differential of a multivariable vector field?

Question

Say I got $\vec r=(xy,y^2,xz)$, can I have the differential $d\vec r$ ?

What would be the generalization of it? I can't find in on internet, so I don't know if it's possible to do.

Answer 1

If $\vec r=(x_1,x_2,x_3)$, where $x_i = x_i (x,y,z)$ . Just take differential of its component functions to get $ d\vec r = (dx_1, dx_2, dx_3)$. That is $$ dx_i = \frac{\partial x_i}{\partial x} dx + \frac{\partial x_i}{\partial y} dy + \frac{\partial x_i}{\partial z} dz $$ So, $$ d\vec r = \sum_{i=1}^3 \Big( \frac{\partial x_i}{\partial x} dx + \frac{\partial x_i}{\partial y} dy + \frac{\partial x_i}{\partial z} dz \Big) \hat{e_i} $$

The generalization of it is obviously a space of covectors of tangent space of manifold. Usually encountered in differential geometry course.