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I've just begun vector calculus course and I've done coordinate systems so far. But I'm confused about deriving basis vectors in coordinate systems.

I don't understand why we can derive standard basis vectors from the derivatives of the location of point p with respect to x, y and z, i.e. I don't understand e$_x$ = $\partial$r/$\partial$$x$ ; e$_y$ = $\partial$r/$\partial$$y$ ; e$_z$ = $\partial$r/$\partial$$z$ .

Similarly, when we find basis unit vectors of coordinates (u, v, w) I don't understand why the basis unit vectors are

e$_u$ $*h_u$ = $\partial$r/$\partial u$

e$_v$ $*h_v$ = $\partial$r/$\partial v$

e$_w$ $*h_w$ = $\partial$r/$\partial w$

(Here, $h_u = $|$\partial$r/$\partial u|$ , $h_v = $|$\partial$r/$\partial v|$ and $h_w = $|$\partial$r/$\partial w|$ )

I don't understand why do we get basis vectors by partial differentiation. Because, I don't think it would be the case in linear algebra.

(Sorry for terrible formatting though. I'm not familiar with writing maths by computer!)

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The fact that they form a basis is reflected in the non-vanishing of the Jacobian at each point $\mathbf u_0$ in parameter-space. That is, $ \det D\mathbf r(\mathbf u_0) \neq 0$ iff $\frac{\partial \mathbf r}{\partial u}(\mathbf u_0),\frac{\partial \mathbf r}{\partial v}(\mathbf u_0),\frac{\partial \mathbf r}{\partial w}(\mathbf u_0)$ are linearly independent.

You need to differentiate to see the relationship with linear algebra because $D\mathbf r(\mathbf u_0)$ is the natural linear map to associate to the coordinates $\mathbf r$ (but it may depend on the point $\mathbf u_0$). In particular if $\mathbf r (x,y,z) = (x,y,z)$ defines cartesian coordinates, then $D\mathbf r = I$ and the columns of $D\mathbf r$ are the 'standard basis' of $\mathbb R^3$.

More generally, for any invertible linear map $L$, the transformation $\mathbf r(\mathbf x) = L\mathbf x$ has derivative $L$. This map sends the standard basis vectors to the columns $\mathbf c_i$ of $L$, and note that $ \frac{\partial \mathbf r}{\partial x_i} = \mathbf c_i$ which matches with change of coordinates from linear algebra.

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