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02.16.08

February 16 roundup of Math 21a

Posted in Math 21a, Spring 2008 at 6:15 pm by leingang

We’re now firmly grounded in three-dimensional space. We talked this week about the various linear objects in 3D: lines, points, planes, and finding equations for them. Also, we showed how to define and calculate the distance between points and lines, lines and planes, etc.

Then we moved into the functional unit of the course. A function of two variables has as its graph a surface in three-dimensional space. Graphing these is a challenge of the brain but of the hands. But there are some tips, such as the method of traces.

Finally, we have other ways to measure space besides our boring old rectangular cartesian coordinates. We defined polar coordinates in the plane, and cylindrical and spherical coordinates in space, to better express certain surfaces. For instance, the cone in three-space can be described as z=\sqrt{x^2+y^2} in cartesian coordinates, but in cylindrical all you need to say is z=r.

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02.07.08

Math 21a roundup for February 6

Posted in Math 21a, Spring 2008 at 2:31 pm by leingang

A vector is a “thing” which has magnitude and direction.  We sort of leave those two things undefined, too, but this is better than nothing.  If you take an abstract algebra course, you’ll learn that a vector is any element of a vector space, but that probably doesn’t help here, either:-). 

Either thinking geometrically (drawing arrows) or algebraically (recording a vector’s components), the important thing is that we can add, subtract, and scale vectors.  Also, these operations are similar to those we know for regular numbers–which we’re going to call scalars from now on for no particularly good reason.  For instance, adding two vectors is a commutative operation: the order of addition doesn’t matter.  Scaling distributes over addition, and so on.  (This is the “abstract” definition of vector space: a set of things you can add and scale).

Can you multiply vectors?  Not in the same sense that you can multiply real numbers.  You can multiply two-dimensional vectors by pretending they’re complex numbers.  This doesn’t work in spaceland, however.  There are two useful products of vectors, though: the scalar (or dot) product and the vector (or cross) product.  The dot product measures the compatibility of vectors in some sense.  If the dot product is positive, the angle between the vectors is acute.  If it’s negative, the angle is obtuse.  And if it’s zero, the vectors are perpendicular (another new word for a known concept: we’ll say orthogonal now instead of perpendicular). 

The cross product is coming up on Friday.

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02.05.08

Math 21a roundup for February 4

Posted in Math 21a, Spring 2008 at 3:39 pm by leingang

First, we played a rousing game of Name Bingo to break the ice. Then we got down to business.

One major goal of Math 21a is to take what we know in the line and on the plane and generalize it to three-dimensional space.

One thing is immediately confusing: we have three axes in space, and we only allow those which are “right-handed” (biased, perhaps, but we have to pick one).

But the distance formula generalizes to three dimensions rather well: the distance between two points is the square root of the sum of the squares of the differences between the original coordinates, or

|(x_1,y_1,z_1)(x_2,y_2,z_2)| = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1 - z_2)^2}

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01.30.08

Happy First Day of Class

Posted in Math 1a, Spring 2008, Math 21a, Spring 2008 at 6:50 am by leingang

Math 21a meets in sections, so it won’t start officially until Monday the 4th.  But Math 1a has its first meetings today and Friday.

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