21 B
Mathematics Math21b Spring 2017
Linear Algebra and Differential Equations
Exhibit: Hidden figures
Course Head: Oliver Knill
Office: SciCtr 432

The math behind Hidden figures

Katherine Johnson at NASA 1966. Image Source
The new movie Hidden figures features some mathematics. One of the entry scenes in elementary school features a quartic equation
    (x2 + 6x - 7) (2x2 - 5 x - 3) = 0
Katherine Coleman Goble Johnson (1918 - ) (played by Lidya Jewett as a kid) factors the polynomial. What does this have to do with linear algebra? First of all we will see that characteristic polynomials of matrices are polynomials. We need to factor them in order to find the eigenvalues. We will also see that if we take a polynomial of differential operators, the factorization allows us to solve differential equations.

The blackboard of Al Harrison (played by Kevin Costner) features equations of conic sections like ellipses given in polar coordinates as
r = α/(1+ε cos(θ)) 
The problem of finding the trajectories of a rocket is a problem in differential equations. One equation we can see
ds/dt = vr,  dvr/dt = vθ2/r g(R/r)
An other describes the motion in spherical coordinates (unfortunately, not all formulas are well visible):
dr/dt    = v sin(y)
dθ/dt    = v cos(y) cos(ψ)/(r cos(φ))  
dφ/dt    = v cos(y) sin(ψ)/r  
dv/dt    = (1/m) Fr - g sin(y) 
v dy/dt  = (1/m) Fm cos(y) -g cos(y) + (v2/r) cos(y) 
v dψ/dt  = (1/m) F...  sin(y)  + (v2/2) 
They are nonlinear but we will cover some nonlinear equations also in our course. Still, the Kepler problem can actually (with some clever transformation) be reduced to a Harmonic oscillator, which is a linear ODE. We see Catherine also work on some differential equations on the large blackboard.

Katherine Johnson, 2015 Source
The scene where the computation of the "go point" for entry is done during the meeting belongs to numerical computations. This is more complicated as there are lots of parameters like thickness of the atmosphere or friction. This is of course done today with the help of computers. In the movie, the IBM 7090 was just introduced. It had a great clock speed of 0.0005 GHz. (As a comparision, a modern phone now computes with 2.33 GHz speed). One also sees the punch card reader IBM 711 and the programming language Fortran (still in use today!). Here are some scenes from the movie which feature math:
Here are some screen shots showing math in the movie
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Math21b Harvard College Course ID:110989| Oliver Knill | Spring 2017 | Department of Mathematics | Faculty of Art and Sciences | Harvard University, [Canvas, for admin], Twitter