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- Summary
- How to interpret a vector differential in the context of differential geometry

For a function ##f: \mathbb{R}^n \to \mathbb{R}##, the following proposition holds:

$$

df = \sum^n \frac{\partial f}{\partial x_i} dx_i

$$

If I understand right, in the theory of manifold ##(df)_p## is interpreted as a cotangent vector, and ##(dx_i)_p## is the basis in the cotangent space at point ##p##, which represents the coordinates map from ##(x_1, \dots, x_n)## to ##x_i##. So the above equation expresses a linear combination. My reference is Prop 4.2 in 'An introduction to manifolds' by Loring W. Tu.

But for a vector differential change ##d\vec{r}##, it seems we get the following result. I've seen this result from different places. One is the Wikipedia article on curvilinear coordinates.

$$

d\vec{r} = \sum^n \frac{\partial \vec{r}}{\partial x_i}dx_i

$$

Here it seems ##d\vec{r}## is not a linear combination of ##dx_i##. Can someone explain these two results and give me a unified view?

$$

df = \sum^n \frac{\partial f}{\partial x_i} dx_i

$$

If I understand right, in the theory of manifold ##(df)_p## is interpreted as a cotangent vector, and ##(dx_i)_p## is the basis in the cotangent space at point ##p##, which represents the coordinates map from ##(x_1, \dots, x_n)## to ##x_i##. So the above equation expresses a linear combination. My reference is Prop 4.2 in 'An introduction to manifolds' by Loring W. Tu.

But for a vector differential change ##d\vec{r}##, it seems we get the following result. I've seen this result from different places. One is the Wikipedia article on curvilinear coordinates.

$$

d\vec{r} = \sum^n \frac{\partial \vec{r}}{\partial x_i}dx_i

$$

Here it seems ##d\vec{r}## is not a linear combination of ##dx_i##. Can someone explain these two results and give me a unified view?