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Ted123
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Homework Statement
[PLAIN]http://img585.imageshack.us/img585/526/indexnotation.jpg [Broken]
The Attempt at a Solution
How do I proceed?
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Hurkyl said:Using index notation sounds like a good place to start...
bigubau said:I would say that r has the components [itex] x,y,z [/itex] or [itex] x_i [/itex]. So the divergence of r is the <scalar product> between the del operator and the r.
So [tex] \mbox{Div} {}\mathbf{r} = (\mathbf{e}_i \partial_i)\cdot (x_j \mathbf{e}_j) [/tex]
So complete the calculation.
Along the same lines you'll solve the 2nd point as well.
hunt_mat said:To start:
[tex]
\mathbf{r}=x^{i}\mathbf{e}_{i}
[/tex]
and div is:
[tex]
\nabla\cdot =\partial_{i}(e_{i}\cdot )
[/tex]
so:
[tex]
\nabla\cdot\mathbf{r}=\sum_{i=1}^{3}\partial_{i}(\mathbf{e}_{i}\cdot (x^{i}\mathbf{e}_{i}))=...
[/tex]
Ted123 said:So how do I evaluate [itex](\mathbf{e}_i \partial_i)\cdot (x_j \mathbf{e}_j) [/itex] ?
Is it just [itex]\mathbf{e}_1 \partial_1 x_1\mathbf{e}_1 + \mathbf{e}_2 \partial_2 x_2\mathbf{e}_2 + \mathbf{e}_3 \partial_3 x_3\mathbf{e}_3[/itex] (this doesn't look right at all)
Ted123 said:Where are these [itex]\mathbf{e}[/itex] vectors coming from? In all my solutions to these questions on index notation I never see an [itex]\mathbf{e}[/itex] appearing.
A similar question to the first one is Show [itex]\nabla (\mathbf{a} \cdot \mathbf{r} ) = \mathbf{a}\;,\;\mathbf{a}\in\mathbb{R}^3[/itex]
and the solution is [itex]\partial_j a_k x_k = a_k \partial_j x_k = a_k \delta_{jk} = a_j[/itex]
Ted123 said:OK so how do I write the 2nd one in terms of the Levi-Civita symbol?
Is it [itex]\varepsilon_{jmn} \partial_m \varepsilon_{nkl} a_k x_l[/itex] ?
If this is right it goes to [itex]a_k ( \delta_{jk} \delta_{ml} - \delta_{jl} \delta_{mk}) \delta_{ml}[/itex]
fzero said:It's less confusing if you use parentheses to keep track of what the derivative acts on:
[itex]\varepsilon_{jmn} \partial_m (\varepsilon_{nkl} a_k x_l)[/itex]
This is true as long as [tex]a_k[/tex] are constants, which is probably intended. Now you should try to compute [tex]\delta_{ml} \delta_{ml} [/tex].
Index notation is a method of writing vector and tensor equations using indices to represent the components of the vectors or tensors. It is also known as Einstein notation or tensor notation.
In index notation, each index represents a specific dimension of a vector or tensor. Repeated indices in an equation indicate summation over those dimensions. This notation allows for a more concise and elegant representation of vector and tensor equations.
Index notation allows for a clearer and more compact representation of vector and tensor equations, making them easier to manipulate and understand. It also helps to avoid errors in calculations and allows for a more elegant and efficient way of solving problems.
Some common operations in vector calculus that use index notation include vector addition, dot product, cross product, and gradient, divergence, and curl operations. These operations can be expressed more succinctly and precisely using index notation.
One of the drawbacks of index notation is that it can be confusing for beginners to understand and use. It also requires a solid understanding of vectors and tensors and their properties. Additionally, some vector calculus operations, such as the determinant, cannot be expressed using index notation.