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values affect where positioned elements sit on that axis; positive values move them higher up the stack, and negative values move them lower down the stack. By default, positioned elements all have a ``` z-index ``` of ``` auto ``` , which is effectively 0.

To change the stacking order, try adding the following declaration to your ``` p:nth-of-type(1) ``` rule:

You should now see the finished example, with the green paragraph on top:

Note that ``` z-index ``` only accepts unitless index values; you can't specify that you want one element to be 23 pixels up the Z-axis — it doesn't work like that. Higher values will go above lower values, and it is up to you what values you use. Using 2 and 3 would give the same effect as 300 and 40000.

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Let's put together a simple example to show what we mean. First of all, delete the existing ``` p:nth-of-type(1) ``` and ``` .positioned ``` rules from your CSS.

Now, update the ``` body ``` rule to remove the ``` position: relative; ``` declaration and add a fixed height, like so:

Now we're going to give the element ``` position: fixed; ``` , and get it to sit at the top center of the viewport. Add the following rule to your CSS:

The ``` top: 0; ``` is required to make it stick to the top of the screen; we then give the heading the same width as the content column and use the faithful old ``` margin: 0 auto; ``` trick to center it. We then give it a white background and some padding, so the content won't be visible underneath it.

If you save and refresh now, you'll see a fun little effect whereby the heading stays fixed, and the content appears to scroll up and disappear underneath it. But we could improve this more — at the moment some of the content starts off underneath the heading. This is because the positioned heading no longer appears in the document flow, so the rest of the content moves up to the top. We need to move it all down a bit; we can do this by setting some top margin on the first paragraph. Add this now:

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Although this product differs from the tensor product, the kernel of Alt is precisely the ideal I (again, assuming that K has characteristic 0), and there is a canonical isomorphism

Suppose that V has finite dimension n , and that a basis e 1 , ..., e n of V is given. then any alternating tensor t ∈ A r ( V ) ⊂ T r ( V ) can be written in index notation as

where t i 1 ⋅⋅⋅ i r is completely antisymmetric in its indices.

The exterior product of two alternating tensors t and s of ranks r and p is given by

The components of this tensor are precisely the skew part of the components of the tensor product s t , denoted by square brackets on the indices:

The interior product may also be described in index notation as follows. Let $t={t}^{{i}_{0}{i}_{1}\cdots {i}_{r-1}}$ be an antisymmetric tensor of rank r . Then, for α V , i α t is an alternating tensor of rank r − 1 , given by

where n is the dimension of V .

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