- #1

mishima

- 570

- 36

- Homework Statement
- Write ##\nabla u## in polar coordinates in terms of its physical components and the unit basis vectors ##e_i##, and in terms of its covariant components and the contravariant basis vectors ##a^i##. What is the relation between the contravariant basis vectors and the unit basis vectors?

- Relevant Equations
- $$V'^i = \frac {\partial x'^i} {\partial x_j} V^j$$

The first part I'm fairly sure is just the regular gradient in polar coordinates typically encountered:

$$\nabla u= \hat {\mathbf e_r} \frac {\partial u} {\partial r} + \hat {\mathbf e_\theta} \frac 1 r \frac {\partial u} {\partial \theta}$$

or in terms of scale factors:

$$=\sum \hat {\mathbf e_i} \frac 1 h_i \frac {\partial u} {\partial x_i}$$

I don't know how to prove that the ##\frac 1 r \frac {\partial u} {\partial \theta}## for example is the so called "physical" component. Are physical components just the dot product of the gradient with each basis?

Similarly I don't know how to show that any particular part, say ##\hat {\mathbf e_\theta} \frac 1 r## is the contravariant basis vector. Do I just plug it into the transformation equation and verify equality$$V'^i = \frac {\partial x'^i} {\partial x_j} V^j ~~~~?$$

$$\nabla u= \hat {\mathbf e_r} \frac {\partial u} {\partial r} + \hat {\mathbf e_\theta} \frac 1 r \frac {\partial u} {\partial \theta}$$

or in terms of scale factors:

$$=\sum \hat {\mathbf e_i} \frac 1 h_i \frac {\partial u} {\partial x_i}$$

I don't know how to prove that the ##\frac 1 r \frac {\partial u} {\partial \theta}## for example is the so called "physical" component. Are physical components just the dot product of the gradient with each basis?

Similarly I don't know how to show that any particular part, say ##\hat {\mathbf e_\theta} \frac 1 r## is the contravariant basis vector. Do I just plug it into the transformation equation and verify equality$$V'^i = \frac {\partial x'^i} {\partial x_j} V^j ~~~~?$$