Index notation and partial derivative

[sanson]

Hi all,

I am having some problems expanding an equation with index notation. The equation is the following:

$$\frac {\partial{u_i}} {dx_j}\frac {\partial{u_i}} {dx_j} $$

I considering if summation index is done over i=1,2,3 and then over j=1,2,3 or ifit does not apply.
Any hint on this would be much appreciated

 

[PeroK]

That's not an equation, that's an expression. I imagine it means:

$$\sum_{i = 1}^{N} \sum_{j = 1}^{N} \frac {\partial{u_i}} {dx_j}\frac {\partial{u_i}} {dx_j} $$ Where the order of the summations is not important.

 

[sanson]

Dear Perok,

Thanks for both, the correction and the explanation. I imagine the same solution but I am not always so sure with the index notation applied to partial derivatives...

to further clarify, that expression gives always an scalar? Or there is not enough information to affirm that?

 

[PeroK]

Dear Perok,

Thanks for both, the correction and the explanation. I imagine the same solution but I am not always so sure with the index notation applied to partial derivatives...

to further clarify, that expression gives always an scalar? Or there is not enough information to affirm that?

It depends how you define a scalar.

 

[sanson]

Let’s say $u$ is the velocity vector and the partial derivative are spatial derivative of the velocity vector. I am wondering if I am getting a single number, a vector or a tensor.

 

[PeroK]

Let’s say $u$ is the velocity vector and the partial derivative are spatial derivative of the velocity vector. I am wondering if I am getting a single number, a vector or a tensor.

A number.

 

[Mark44]

Hi all,

I am having some problems expanding an equation with index notation. The equation is the following:

$$\frac {\partial {u_i}} {dx_j}\frac {\partial{u_i}} {dx_j} $$

I considering if summation index is done over i=1,2,3 and then over j=1,2,3 or ifit does not apply.
Any hint on this would be much appreciated

With the corrections shown below, the expression above looks to me like it could also be written as $$\left(\frac{\partial u_i} {\partial x_j} \right)^2$$
Note that in partial derivatives you don't mix the partial derivative symbol $\partial$ with the $d$ used in ordinary derivatives.

OTOH, if the intent was to write a 2nd partial derivative, it could be written like this:
$$\frac {\partial}{ \partial x_j} \left(\frac {\partial u_i} {\partial x_j}\right) $$