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I'm trying to learn how to do vector calculus in Mathematica, just seem to be missing some points along the learning curve.

With assumptions

$Assumptions = {
  (r[t] | r'[t] | r''[t]) \[Element] Vectors[{n}, Reals],
  n >= 2}

I first use sigma[r[t]]==0 to define a surface, I then want to calculate the first and second time derivatives of this(which should both be equal to zero, but I need the algebraic equations for what I'm doing). Evaluating

D[sigma[r[t]], {t}]
D[sigma[r[t]], {t, 2}]

I should get(copy/pasting the following into mathematica should give the correct symbols)

r'[t] . Gradient[sigma[r[t]]]
r''[t] . Gradient[sigma[r[t]]] + (r'[t])^2 . (\[Del]*Gradient[sigma[r[t]]])

where * is the Hadamard product. But I'm instead getting

Derivative[1][r][t] Derivative[1][sigma][r[t]]
Derivative[1][sigma][r[t]] (r^\[Prime]\[Prime])[t] + 
 Derivative[1][r][t]^2 (sigma^\[Prime]\[Prime])[r[t]]

What should I be doing differently to get my equations in the correct form? This is very close, but the lack of dot products doesn't allow calculations to proceed as expected.

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  • $\begingroup$ Mathematica can't do this kind of vector calculus out of the box in a component-free way. You need to break things up into components like D[sigma[{rx[t], ry[t]}], t] $\endgroup$
    – flinty
    Commented Feb 3 at 11:19

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