Notation Help for Derivatives in Continuum Mechanics

  • Thread starter J Hill
  • Start date
  • Tags
    Notation
In summary, the conversation discusses tensor derivatives in continuum mechanics and the notation for these equations. The first equation, with a scalar field f, involves the gradient of f and the directional derivative in the direction of u. The second equation, with a vector field f, also involves the gradient but is represented as a second order tensor in indicial notation. The dot product is seen as a contraction and the result of the operation is a scalar.
  • #1
J Hill
12
0
I've recently begun to study continuum mechanics, and am having difficulty with tensor derivatives, primarilly because of notation... so can anyone explain what these equations mean, as far as notation goes:
1. [tex]\frac{\partial f}{\partial\mathbf{v}}\cdot\mathbf{u}=Df(\mathbf{v})[\mathbf{u}][/tex]
2. [tex]\frac{\partial \mathbf{f}}{\partial \mathbf{v}}\cdot\mathbf{u}=D\mathbf{f}(\mathbf{v})[\mathbf{u}][/tex]
 
Physics news on Phys.org
  • #2
Hi J,

[tex]
\frac{\partial \mathbf{f}}{\partial \mathbf{v}}
[/tex]

is a second order tensor: it is the gradient of the vector field f, which in cartesian coordinates would be represented by the Jacobian matrix of the function f as a vector valued function of the vairables vi ,
f=(f1(v1,v2,...,vn),f2(v1,v2,...,vn),...,fn(v1,v2,...,vn))
(though please note that

[tex]
\frac{\partial \mathbf{f}}{\partial \mathbf{v}}
[/tex]

is a tensor, not a matrix). I prefer to write it as:

[itex]
\nabla \mathbf{f}
[/itex]

as this way it's coordinate independent.
The dot product is seen as a contraction, so that

[itex]
\nabla \mathbf{f} \cdot\mathbf{u}
[/itex]

yields a vector (or first order tensor), which is basically the directional derivative of the field f in the direction of u (or proportional to it if u has modulus different to 1).

I don't like the notation
[tex]
D\mathbf{f}(\mathbf{v})[\mathbf{u}]
[/tex]
because this suggests that
[tex]
D\mathbf{f}(\mathbf{v})
[/tex]
and
[tex]
[\mathbf{u}]
[/tex]
are matrices rather than tensors (a square matrix and a column matrix respectively), and all you have to do is the matrix multiplication. Oh well as long as you don't forget that the matrices just represent the components of the tensors in a certain basis :p

My favourite way of writing

[itex]
\nabla \mathbf{f}
[/itex]

though, is using indicial notation, because it leaves no room for ambiguity. Thus,

[itex]
\nabla \mathbf{f}
[/itex]
is written as:
[itex]
\mathbf{e_i}\frac{\partial\mathbf{f}}{\partial v_i} = \mathbf{e_i}\frac{\partial f^j \mathbf{e_j}}{\partial v^i}
[/itex]
so that

[itex]
\nabla \mathbf{f} \cdot\mathbf{u} = \mathbf{e_i}\frac{\partial\mathbf{f}}{\partial v_i} \cdot\mathbf{u}
= \mathbf{e_i}\frac{\partial f^j \mathbf{e_j}}{\partial v^i} \cdot u^k \mathbf{e_k}
[/itex]

In a Cartesian coordinate system this would equal:
[itex]
\nabla \mathbf{f} \cdot\mathbf{u} =
= \mathbf{e_i}\frac{\partial f_j \mathbf{e_j}}{\partial v^i} \cdot u_k \mathbf{e_k} = \mathbf{e_i}\frac{\partial f_j }{\partial v_i} \mathbf{e_j} \cdot u_k \mathbf{e_k} =
\mathbf{e_i}\frac{\partial f_j }{\partial v_i} u_k (\mathbf{e_j} \cdot \mathbf{e_k}) =
\mathbf{e_i}\frac{\partial f_j }{\partial v_i} u_k \delta_j_k =
\mathbf{e_i}\frac{\partial f_j }{\partial v_i} u_j
[/itex]

Hope this helps! :)
 
  • #3
Oops, I showed you what 2. is only, 1. is the same except that f is a scalar field, ie. same as the second case but with just one component, so that grad(f) is a vector and the result of your operation is a scalar:
[itex]

\nabla f \cdot\mathbf{u}
= \mathbf{e_i}\frac{\partial f }{\partial v^i} \cdot u_k \mathbf{e_k} =
\frac{\partial f}{\partial v_i} u_k (\mathbf{e_i} \cdot \mathbf{e_k}) =
\frac{\partial f}{\partial v_i} u_k \delta_i_k =
\frac{\partial f}{\partial v_i} u_i

[/itex]
 

Related to Notation Help for Derivatives in Continuum Mechanics

1. What is the notation used for derivatives in continuum mechanics?

The most commonly used notation for derivatives in continuum mechanics is the partial derivative symbol, denoted by , followed by the variable with respect to which the derivative is being taken. For example, ∂u/∂x represents the partial derivative of the function u with respect to x.

2. How do you represent higher order derivatives in continuum mechanics?

Higher order derivatives in continuum mechanics are represented by adding additional symbols. For example, ∂2u/∂x2 represents the second derivative of the function u with respect to x.

3. What is the difference between a partial derivative and a total derivative?

A partial derivative is taken with respect to one variable while holding all other variables constant. In contrast, a total derivative takes into account changes in all variables. In continuum mechanics, total derivatives are often represented by the symbol d.

4. How do you denote derivatives with respect to time in continuum mechanics?

Derivatives with respect to time in continuum mechanics are typically denoted by a dot above the variable. For example, represents the time derivative of the function u.

5. Are there any alternative notations for derivatives in continuum mechanics?

Yes, there are alternative notations for derivatives in continuum mechanics, such as the use of subscripts instead of symbols, or the use of primes instead of dots for time derivatives. However, the and dot notations are the most commonly used and widely accepted in the field.

Similar threads

Replies
3
Views
1K
Replies
13
Views
1K
Replies
4
Views
529
Replies
1
Views
4K
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
968
Replies
4
Views
3K
  • Special and General Relativity
2
Replies
47
Views
3K
  • Calculus
Replies
4
Views
937
  • Advanced Physics Homework Help
Replies
5
Views
2K
Back
Top