Chapter 10: Q5P (page 534)

Write $\nabla u$ in polar coordinates in terms of its physical components and the unitbasis vectors $e_{i}$ , and in terms of its covariant components and the contravariantbasis vectors $a_{i}$ . What is the relation between the contravariant basis vectors andthe unit basis vectors? Hint:Compare equation (10.11) and our discussion of it.

Short Answer

Expert verified

The equation a gradient in terms of its unit basis vectors.

$\nabla u = \frac{\partial u}{\partial r} {\hat{e}}_{r} + \frac{1}{r} \frac{\partial u}{\partial θ} {\hat{e}}_{θ}$

The equation of a gradient in terms of its contravariant components.

$\nabla u = \frac{\partial u}{\partial r} a^{r} + \frac{\partial u}{\partial θ} a^{θ}$

Step by step solution

Given information.

Unit basis vectors and contravariant basis vectors are given.

Definition of a covariance and contravariance.

The components of a vector relative to a tangent bundle basis are covariant in differential geometry if they change with the same linear transformation as the basis. If they change as a result of the inverse transformation, they are contravariant.

Define a gradient.

Define a gradient as calculated in a polar coordinate system.

$\nabla u = \frac{\partial u}{\partial r} {\hat{e}}_{r} + \frac{1}{r} \frac{\partial u}{\partial θ} {\hat{e}}_{θ}$

Express it in terms of contravariant basis vectors.

Express vectorsin terms of contravariant vectors. Make use of the metric tensor.

$a^{i} = g^{i j} a_{j}$

$\begin{matrix} a^{r} = \frac{1}{h_{r}^{2}} a_{r} \\ = \frac{1}{h_{r}} {\hat{e}}_{r} \\ = {\hat{e}}_{r} \end{matrix}$

Continue for the next component.

$\begin{matrix} a^{θ} = \frac{1}{h_{θ}^{2}} α_{θ} \\ = \frac{1}{h_{θ}} {\hat{e}}_{θ} \\ = \frac{1}{r} {\hat{e}}_{θ} \end{matrix}$

Consolidate the previous equations to write the equation of a gradient.

Combine these equations and write the equation of a gradient.

$\nabla u = \frac{\partial u}{\partial r} a^{r} + \frac{\partial u}{\partial θ} a^{θ}$

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Short Answer

Step by step solution

Given information.

Definition of a covariance and contravariance.

Define a gradient.

Express it in terms of contravariant basis vectors.

Consolidate the previous equations to write the equation of a gradient.

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