4605.sagews

x,y=var('x,y')
f(x,y)=(exp(x)*cos(y),exp(x)*sin(y))
show(jacobian(f, [x,y]))

\newcommand{\Bold}[1]{\mathbf{#1}}\left($$\begin{array}{rr} \left( x, y \right) \ {\mapsto} \ e^{x} \cos\left(y\right) & \left( x, y \right) \ {\mapsto} \ -e^{x} \sin\left(y\right) \\ \left( x, y \right) \ {\mapsto} \ e^{x} \sin\left(y\right) & \left( x, y \right) \ {\mapsto} \ e^{x} \cos\left(y\right) \end{array}$$\right)

show(jacobian(f,[x,y]).det().full_simplify())

\newcommand{\Bold}[1]{\mathbf{#1}}\left( x, y \right) \ {\mapsto} \ e^{\left(2 \, x\right)}

stuff = [line(L,rgbcolor=(1,1/4,1/2)) for L in [[f(a,b*pi/20) for a in range(-5,4)] for b in range(-21,20)]]
new_plot=stuff[0]
for j in stuff[1:]:
    new_plot+=j
stuff = [line(L,rgbcolor=(0,1/4,1/2)) for L in [[f(a,b*pi/20) for b in range(-21,20)] for a in range(-5,4)]]
for j in stuff:
    new_plot+=j
new_plot.show()