SageMath syntax for Section 4.6

SageMath is a free open-source mathematics software system. You can use a SageMath cell online by going here.

• Gradient of $f(x,y)=xe^{-3y}+\sin(2x-5y)$

x
var("x,y")
f = x*exp(-3*y) + sin(2*x-5*y)
grad = vector([diff(f,x),diff(f,y)])
grad

• Directional derivative of $f(x,y)=xe^{-3y}+\sin(2x-5y)$ at $(0,0)$ in the direction of $\langle 1,2 \rangle$

x
var("x,y")
f = x*exp(-3*y) + sin(2*x-5*y)
grad = vector([diff(f,x),diff(f,y)])
v=vector([1,2])
grad(x=0,y=0).dot_product(v.normalized())

• Gradient of $g(x,y,z)=x^2+y^3+z^4$

xxxxxxxxxx
var("x,y,z")
g = x^2+y^3+z^4
grad = vector([diff(g,x),diff(g,y),diff(g,z)])
grad