difference inner, dot

Question

Is there a difference between

dot(u, v)

and

inner(u, v)

for vector-valued functions u, v?

Answer 1

Yes, inner(u, v)$=u_{ijk\ldots}v_{ijk\ldots}$, dot(u, v)$=u_{ijk\ldots n}v_{nmo\ldots}$ (summation implied).

Now I read "$u$, $v$ vector-valued", then answer is no:)

Comments

Can you give an trivial example about this?

---

V = VectorFunctionSpace(mesh, 'CG', 2)
Q = FunctionSpace(mesh, 'CG', 1)

u = Function(V) # velocity
p = Function(Q) # pressure

I = Identity(V.cell().d)
T = -p*I + 2.0*mu*sym(grad(u))) # Cauchy stress (for isochoric motion of Newtonian fluid)

traction = dot(T, n) # for n unit normal of respective surface

stress_power = inner(T, grad(u))

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When u, v are vectors (only have one index ) not tensor.
inner(u , v) = u_i v_i
dot(u,v) = u_i v_i
Does than mean they are the same or my understanding was wrong?
Thanks :)

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Does than mean they are the same or my understanding was wrong?

Sure.

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The are the same for first order tensors, but not for second order tensors.

inner(u_ij, v_kl) = u_ij v_ij
dot(u_ij, v_k) = u_ij v_jk