Exercise [12.05]

Sameed Zahoor · 30 Aug 2008, 11:13

Let,
$\alpha=\alpha_1 dx^1 +\alpha_2 dx^2+...+\alpha_{n} dx^{n}$
be a general covector field.
Then the covector dx^2

is a covector field with a certain set of components $\alpha_{i}$ .The components can be determined by equating $dx^{2}$ to $\alpha$ .
Hence,
$dx^2=\alpha_1 dx^1 +\alpha_2 dx^2+...+\alpha_{n} dx^{n}$
As long as $dx^{i}$ and $dx^{j}$ are linearly independent for

not equal to

,the components $(\alpha_1,\alpha_2,...,\alpha_{n})$ can be seen to be (0,1,0,...,0)

by simply equating the coefficients of $dx^{i}$ on both sides.

Exercise [12.05]

Who is online