- #1

- 12

- 0

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]

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- Thread starter J Hill
- Start date

- #1

- 12

- 0

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]

- #2

- 162

- 0

[tex]

\frac{\partial \mathbf{f}}{\partial \mathbf{v}}

[/tex]

is a second order tensor: it is the gradient of the vector field

(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

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

- 162

- 0

[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]

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