Tutorial 2

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Topics Covered

This tutorial covers the following topics using Python and SymPy:

• Partial Derivatives: Using limit definition and rules.
• Chain Rule: For composite functions.
• Implicit Differentiation: For multivariable functions.
• Jacobians: For transformations.

Introduction

Sympy: SymPy is a Python library for symbolic mathematics. We will use this library to solve all the tutorials.

import sympy as sy
from sympy.abc import x, y, z, s, t, r, theta, u, v # variables
from sympy.solvers import solve                   # Solving equations
from sympy import sqrt,tan,sin,cos,sec,pi,root,ln, E
from sympy import log,exp,atan,asinh,atanh,asin   # Import all required math function
from sympy import diff, idiff, Matrix                     # Solving differentiation
sy.init_printing(use_latex=True)                  # Show it in natural display

Question 1

Find the partial derivative $\displaystyle\left(\frac{\partial f}{\partial y}\right)$ and $\displaystyle\left(\frac{\partial f}{\partial x}\right)$ of these functions using the limit definition

a. $f(x, y)=x^{2} y+2 x+y^{3}$

f = x**2*y + 2*x + y**3
display(diff(f, x))
display(diff(f, y))

b. $f(x, y)=x^{2}-4 x y+y^{2}$

f = x**2 - 4*x*y + y**2
display(diff(f, x))
display(diff(f, y))

c. $f(x, y)=2 x^{3}+3 x y-y^{2}$

f = 2*x**3 + 3*x*y - y**2
display(diff(f, x))
display(diff(f, y))

Question 2

Determine all the first and second order partial derivatives of the function

a. $f(x, y)=x^{2} y^{3}+3 y+x$

f = x**2*y**3 + 3*y + x
display(diff(f, x))
display(diff(f, y))
display(diff(f, x, x))
display(diff(f, x, y))
display(diff(f, y, x))
display(diff(f, y, y))

b. $f(x, y)=x^{4} \sin 3 y$

f = x**4*sin(3*y)
display(diff(f, x))
display(diff(f, y))
display(diff(f, x, x))
display(diff(f, x, y))
display(diff(f, y, x))
display(diff(f, y, y))

c. $f(x, y)=x^{2} y+\ln \left(y^{2}-x\right)$

f = x**2*y + ln(y**2 - x)
display(diff(f, x))
display(diff(f, y))
display(diff(f, x, x))
display(diff(f, x, y))
display(diff(f, y, x))
display(diff(f, y, y))

d. $f(x, y)=e^{x y}(2 x-y)$

f = exp(x*y)*(2*x - y)
display(diff(f, x))
display(diff(f, y))
display(diff(f, x, x))
display(diff(f, x, y))
display(diff(f, y, x))
display(diff(f, y, y))

Question 3

Find both partial derivatives for each of the following two variables functions

a. $g(x, y)=y e^{x+y}$

g = y*exp(x+y)
display(diff(g, x))
display(diff(g, y))

b. $h(x, y)=x \sin y-y \cos x$

h = x*sin(y) - y*cos(x)
display(diff(h, x))
display(diff(h, y))

c. $p(x, y)=x^{y}+y^{2}$

p = x**y + y**2
display(diff(p, x))
display(diff(p, y))

d. $U(x, y)=\frac{9 y^{3}}{x-y}$

U = (9*y**3)/(x-y)
display(diff(U, x))
display(diff(U, y))

Question 4

For $f(x, y, z)$, use the implicit function theorem to find $d y / d x$ and $d y / d z$

a. $f(x, y, z)=x^{2} y^{3}+z^{2}+x y z$

f = x**2*y**3 + z**2 + x*y*z
dydx = -diff(f, x) / diff(f, y)
dydz = -diff(f, z) / diff(f, y)
display(dydx)
display(dydz)

b. $f(x, y, z)=x^{3} z^{2}+y^{3}+4 x y z$

f = x**3*z**2 + y**3 + 4*x*y*z
dydx = -diff(f, x) / diff(f, y)
dydz = -diff(f, z) / diff(f, y)
display(dydx)
display(dydz)

c. $f(x, y, z)=3 x^{2} y^{3}+x z^{2} y^{2}+y^{3} z x^{4}+y^{2} z$

f = 3*x**2*y**3 + x*z**2*y**2 + y**3*z*x**4 + y**2*z
dydx = -diff(f, x) / diff(f, y)
dydz = -diff(f, z) / diff(f, y)
display(dydx)
display(dydz)

Question 5

Find $\frac{\partial F}{\partial s}$ and $\frac{\partial F}{\partial t}$, if applicable, for the following composite functions

a. $F=\sin (x+y)$ where $x=2st$ and $y=s^2+t^2$

F = sin(x+y)
x_s = 2*s*t
y_s = s**2 + t**2
F_s = F.subs([(x, x_s), (y, y_s)])
display(diff(F_s, s))
display(diff(F_s, t))

b. $F=\ln \left(x^{2}+y\right)$ where $x=\mathrm{e}^{(s+t 2)}$ and $y=s^{2}+t$

F = ln(x**2 + y)
x_st = exp(s+t**2)
y_st = s**2 + t
F_st = F.subs([(x, x_st), (y, y_st)])
display(diff(F_st, s))
display(diff(F_st, t))

c. $F=x^{2} y^{2}$ where $x=s \cos t$ and $y=s \sin t$

F = x**2*y**2
x_st = s*cos(t)
y_st = s*sin(t)
F_st = F.subs([(x, x_st), (y, y_st)])
display(diff(F_st, s))
display(diff(F_st, t))

d. $F=xy+y \mathrm{z}^{2}$ where $x=\mathrm{e}^{t}, y=\mathrm{e}^{t} \sin t$ and $\mathrm{z}=\mathrm{e}^{t} \cos t$

F = x*y + y*z**2
x_t = E**t
y_t = E**t*sin(t)
z_t = E**t*cos(t)
F_t = F.subs([(x, x_t), (y, y_t), (z, z_t)])
display(diff(F_t, t))

Question 6

Find $d y / d x$ and $d y / d z$ (if applicable) for each of the following

a. $7 x^{2}+2 x y^{2}+9 y^{4}=0$

f = 7*x**2 + 2*x*y**2 + 9*y**4
dydx = idiff(f, y, x)
display(dydx)

b. $x^{3} z^{2}+y^{3}+4 x y z=0$

f = x**3*z**2 + y**3 + 4*x*y*z
dydx = idiff(f, y, x)
dydz = idiff(f, y, z)
display(dydx)
display(dydz)

c. $3 x^{2} y^{3}+x z^{2} y^{2}+y^{3} z x^{4}+y^{2} z=0$

f = 3*x**2*y**3 + x*z**2*y**2 + y**3*z*x**4 + y**2*z
dydx = idiff(f, y, x)
dydz = idiff(f, y, z)
display(dydx)
display(dydz)

d. $y^{5}+x^{2} y^{3}=1+y \exp \left(x^{2}\right)$

f = y**5 + x**2*y**3 - (1 + y*exp(x**2))
dydx = idiff(f, y, x)
display(dydx)

Question 7

a. In polar coordinates, $x=r \cos \theta, y=r \sin \theta$, show that $\frac{\partial(x, y)}{\partial(r, \theta)}=r$

X = Matrix([r*cos(theta), r*sin(theta)])
Y = Matrix([r, theta])
J = X.jacobian(Y)
display(J)
display(J.det())

b. Obtain the Jacobian $J$ of the transformation $s=2 x+y, t=x-2 y$ and determine the inverse of the transformation $J_{1}$. Confirm that $J_{1}=J^{-1}$.

S = Matrix([2*x+y, x-2*y])
X = Matrix([x, y])
J = S.jacobian(X)
display(J)
display(J.det())

# Inverse
sol = solve([s - (2*x+y), t - (x-2*y)], (x, y))
x_st = sol[x]
y_st = sol[y]
X_st = Matrix([x_st, y_st])
S_st = Matrix([s, t])
J1 = X_st.jacobian(S_st)
display(J1)
display(J1.det())
display(J.inv())

c. Show that if $x+y=\mathrm{u}$ and $y=\mathrm{uv}$, then $\frac{\partial(x, y)}{\partial(u, v)}=u$.

# x = u - y => x = u - uv
x_uv = u - u*v
y_uv = u*v
X = Matrix([x_uv, y_uv])
Y = Matrix([u, v])
J = X.jacobian(Y)
display(J)
display(J.det())

d. Verify whether the functions $u=\frac{x+y}{1-x y}$ and $v=\tan ^{-1} x+\tan ^{-1} y$ are functionally dependent.

u_xy = (x+y)/(1-x*y)
v_xy = atan(x) + atan(y)
U = Matrix([u_xy, v_xy])
X = Matrix([x, y])
J = U.jacobian(X)
display(J)
display(J.det().simplify())

The Jacobian is 0, so the functions are functionally dependent.

e. If $x=uv, y=\frac{u+v}{u-v}$, find $\frac{\partial(u, v)}{\partial(x, y)}$.

jacobian = Matrix([[diff(u*v, u), diff(u*v, v)],
                   [diff((u+v)/(u-v), u), diff((u+v)/(u-v), v)]])
simplify(jacobian.det())  # ∂(x,y)/∂(u,v)