Wolfram Alpha, MATLAB, Calculators, and all the other computing technologies
Once we could calculate the inverse of a 4x4 matrix or complete triple integration by parts by hand. Now, why bother?
What do you do when your professor tells you to stop learning maths because you will never be in a situation in the future where you were restricted access to a computer?
Remote learning and the educational implications of COVID-19 aside, most students at institutions of higher education fortunate to have unlimited access to very fast, accurate, and powerful computing technologies. Some teachers and professors have adopted the stance that, therefore, it is no longer necessary to instruct in the methods of “grinding through the algebra”. Instead, they grant their students the ability to access anything and everything they could imagine during an exam and focus on teaching math in a new, futuristic, and either quite depressing or quite empowering way (depending on how you see it).
Before this year I didn’t even realise what I was missing out on with softwares like Wolfram Alpha, Desmos, Geogebra, MATLAB, Symbolab, the list goes on. These sites can perform eigen analysis one second, and tell a bad physics joke the next. You can plot in several dimensions, differentiate or integrate until you turn blue, animate transformations, and calculate 2+2.
Does this open up a whole new world of endless potential for mathematical feats, or make us lazy, mind-numb f**** who will eventually lose our foundations in mathematics? Should we waste time memorising “ulta violet voodoo” and “low d-high minus high d-low all over low squared”, or should we leave it to the technology that these past geniuses so gracefully left for us? I am not here to answer this, but ask you!
WOLFRAM
Functions
Plot multiple functions on the same graph
plot y=e^-x and y=cos(pi x) and y=cos(pi x) e^-x for 0<x<5
Determine values of a function over specific domain and range
y=(x^2-3x+2) and y=(2x-5) and y=(x^2-3x+2)/(2x-5) for 0<x<4 and -3<y<3
Quickly find asymptotes of a function
asymptotes y=(x^2-3x+2)/(2x-5)
Test values of a function that contains an unknown variable and plot all versions
plot y=5x^2-3ax-a where a=(1,2,3,4)
Change the axes to semi-log
log plot y=2^x +2 and y=2^(x+2) for -3<x<3
Vectors
Simply plot vectors in 3D space to visualise
vector 2i - 4j + 2k and 3i +3j
Add vectors using different kind of notation
vector (2i - 4j + 2k) + (3i +3j) OR vector (2,-4,2) + (3,3,0)
Dot product
(3i-2j+4k).(5i-4j-2k)
Cross product
(3i-2j+4k)x(5i-4j-2k)
Triple scalar product
(3i-2j+4k)x(5i-4j-2k).(2i-2j-8k)
Matrices
Add matrices
{{2, 2}, {3, 8}} + {{-2, 3}, {1, 5}}
Scalar multiplication
3.{{2, 2}, {3, 8}}
Multiply matrices
{{2, 2}, {3, 8}} . {{-2, 3}, {1, 5}}
Find the determinant
det ({{2, 1, 2}, {-2, -3, 1}, {-1, -2, 1}})
Find the inverse of a square matrix
inv ({{2, 1, 2}, {-2, -3, 1}, {-1, -2, 1}})
Solve linear systems of equations in matrix form
{{2, 2, 3}, {3, 8, 2}, {-1, 2, -1}} .{x, y, z}={20, 50, 30}
Linear Transformations
Rotations
{{cos(a), -sin(a)}, {sin(a), cos(a)}} . {1, 1} where a=90 degree
Shears
{{1, a}, {0, 1}} . {1, 1} where a=2
3D scaling
{{a, 0, 0}, {0, b, 0},{0, 0, c}} . {2, 2, 2} where a=2, b=1, c=3
Solving non-homogeneous differential equations
{{2, 6},{1, 2}}.{x, y}={7, 5}
Visualising simultaneous equations
2x + 6y=7 and 1x + 2y =5
Eigenvalues
Find eigenvalues and corresponding eigenvectors
eigenvalues {{4,1},{2,-1}}
Complex Numbers
Find different forms of complex number by just typing in rectangular form
4-3i
Computations with complex numbers
(2+3i)^3/(5-i)^2
Calculate all the roots of a particular equation
z=(1+i)^(1/5)
Series
Use series summing calculator
sum
Expand approximation about a certain point
taylor series sin x at x=pi/2
Specify the order of approximation
series (sin x)/(x-pi) at x=pi to order 10
Inspect convergence of a particular sum
sum convergence(1/x^n,n)
ODEs
Visualise functions and their derivatives on the same plot
plot exp(x) and exp(-x) and derivative exp(x) and derivative exp(-x)
Plot trigonometric functions and their derivatives
plot sin(x) and cos(x) and derivative sin(x) and derivative cos(x)
Solve spring-mass system equations given initial conditions
5x'' + 2x = 0, x(0)=0.5 and x'(0)=0
Solve spring-mass-damper equations given initial conditions
5x'' +1x'+ 2x = 0, x(0)=0.5 and x'(0)=0
** Piecewise on Wolfram:
Piecewise[{{-t, -pi < t < 0}, {t, 0<t<pi}}]
** Piecewise on demos
f\left(t\right)=\left\{-\pi<t<0:\ -t,\ 0<t<\pi:\ t\right\}
MATLAB
Finding eigenvalues
eig(A)
Returns diagonal matrix D of eigenvalues and a matrix V whose columns are the corresponding eigenvectors
[V,D,] = eig(A)
