Mathematical Journeys - Saint Ann's School

39
LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS
THEO MCKENZIE
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This year for my independent math research I explored the fields of linear
algebra and multivariable calculus. These two subjects in mathematics are both highly
applicable and are usually learnt in the first year of college. Linear algebra is the study
of linear spaces, i.e. spaces defined by vectors. Multivariable calculus is used to
describe and analyze equations defined by more than one parameter.
After learning basic rules of linear equations I applied this to some simple
problems. One of the first problems that I did was the examination at a methane
molecule. In methane (CH 4 ) the electrons of hydrogen are attracted towards the
carbon atom, giving each of the hydrogen atoms a positive charge. This makes them
repel each other and go as far away as possible from each other. This creates a regular
tetrahedron with the carbon atom in the center. The problem was to find the angle
between two hydrogen atoms. This was doable because I could use an equality
equation with the dot product and solved for the angle.
Another problem to do with linear equations was finding the area of the face
of a tetrahedron, i.e. a triangle. This turns out to be double the cross product of any of
the two vectors compromising the triangle, considering that the cross product by
definition is the magnitude of two vectors multiplied by the sine of the angles between
them. This works out to the base of a triangle multiplied by the height, which when
divided by two is the area of a triangle.
I then looked at matrices and their function in linear spaces. Various matrices
have interesting properties, such as a matrix that rotates a vector a given angle in two
dimensions, matrix inverses and orthogonal matrices. Matrices can also be used to
solve linear equations. One problem that I did relating to this field was related to
cooking. I was able to find out the nutritional value of various pastries from two given
matrices in a small amount of time by putting the ingredients and the nutritional
values of the ingredients into matrices, then multiplying them using the inner product.
In terms of problems to do with multivariable calculus, I began by learning the
basic rules of differentiation of functions of more than one variable. I had already
made myself familiar with partial derivatives so most of my work was with the del in
terms of the gradient, the divergent and the curl. I used these to describe many graphs,
finding the maxima and minima and the tangent plane. Probably the problem that I
spent the most time on this year was to find the region defined by
0 < ∫∫ sin( x + y)dA