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Calculus: Part Three - Vector Calculus


How to find us:

You can find the Engineering Library, across the street from the Husky Union Building. Make sure to take time to know how to find the building since the UW campus is big. The easiest way is to come in on Memorial Way near the Burke Museum and then make an immediate right to walk on to E Stevens Way NE (please look at the map (http://www.washington.edu/maps/?l=ELB). We will be in one of the conference rooms on the second to fourth floor and will post a sign out saying "Math Group". It's not a big library so it should be easy to find us once you find the building.


Come learn vector calculus! Vector calculus is an essential tool in applied mathematics. This is the second part in our calculus series. Some of the topics we will cover in this talk include:

• A very brief overview of vectors

• The intuition for vector calculus operations

• Gradient

• Curl

• Divergence

• Laplacian

Again, the chief goal will be to enable you to walk away with the ability to intuitively read vector calculus equations and compute vector calculus operations. 



Picture from Wikipedia

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  • Danny C

    Damn snow

    1 · February 10, 2014

    • Nile

      I hear you. I honestly thought there was a good chance I would be talking to an empty room :].

      February 10, 2014

  • Jon C.

    Another way to look at cross product: When take (x,y) and transform it to (y,-x) you rotate it by 90 degrees counterclockwise. So (x1*y2-x2*y1) is the same as rotating (x2,y2) and then taking the dot product with (x1,y1). Since dot product measures parallelness, each term of the cross product measures parallelness after rotating the vector in the corresponding plane.

    1 · February 9, 2014

    • Nile

      I like this. Very nice illustration of the geometric relationship between the two. On another note, if there had been more time, I would have liked to derive the geometric dot product and cross-product from their algebraic equations (pretty simple algebraic proofs, see for example http://www.proofwiki....­).

      February 10, 2014

  • Jon C.

    Here's how I see the curl: for the X coordinate, we look at the contribution to the swirl caused by changes in y and x. If you move by dy and notice Fz increase, then that is counter-clockwise swirl. If you move by dz and notice Fy increase, then that is clockwise swirl. Subtract the two and it is the net swirl at that point around the i vector.

    1 · February 9, 2014

    • Nile

      Yes! Take V=zj + yk. If you graph this and look the vectors that are produced it's easy to see that the clockwise and counter-clockwise swirls are canceling each other out leading to no curl in the x-direction. This is easy to graph as well, would have been a good example.

      February 10, 2014

  • Nile

    References:

    (best by a mile)
    A Student's Guide to Vectors and Tensors

    (decent)
    Introduction to Electrodynamics by David J. Griffiths

    February 7, 2014

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