we find that the z component of ru x rv =

v^2u^2 – 4(u^2v^2)

which they calculate as -2(u^2v^2) instead of -3(u^2v^2)

]]>Alternatively, you can also say that if F is incompressible (has divergence zero), and it is defined in the region between the two surfaces, then the statement of equality is true. This uses the Divergence Theorem.

]]>If F is the curl of another vector field, and C is the same boundary for S1 and S2 then according to Stokes’ theorem the original statement is true.

….in the right ball park?

]]>Not the way we’ve defined it, no. The surface area is an integral over a region of a non-negative quantity, .

The given Klein bottle has a well-defined finite positive surface area (depending on what size and shape Klein bottle it is!). By surface area I mean the amount of glass (in square centimetres, say) that you would use to construct it. In other words, the number of sides doesn’t matter to the surface area computation (we don’t count each square centimetre twice or anything like that).

The number of sides *does* matter when we start talking about surface integrals of a vector field.

]]>I was wondering if anyone got 15% or higher on their 2nd midterm if you would be willing to take the 2nd midterm mark in place of both midterms.

This happened in my linear algebra class, and it provided a lot of motivation, especially if you didn’t poorly on the first midterm to hop on the band wagon and do really well on the 2nd midterm.

Just a suggestion.

Thanks!

]]>Thank you, that was my intention. I’ve edited the post to make this clear.

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