Partial Differentiation

Let \(z = f(x, y)\) be a function of two independent variables \(x\) and \(y\), then the derivative of \(z\) with respect to \(x\) keeping \(y\) constant is called the partial derivate of \(z\) with respect to \(x\)

Notation

\[ \frac{\partial f}{\partial x} = f_x, \quad \frac{\partial f}{\partial x} = f_y, \quad \frac{\partial^2 f}{\partial x^2} = f_{xx}, \quad \frac{\partial }{\partial x} \left( \frac{\partial f}{\partial y} \right) = f_{xy}, \quad \frac{\partial }{\partial y} \left( \frac{\partial f}{\partial x} \right) = f_{yx}, \quad \frac{\partial^2 f}{\partial y^2} = f_{yy} \]

Limit of function with two variables

• limit of function with two variables
  • a function \(f(x, y)\) is said to tend to limit \(L\) as
  • \(x \rightarrow a\) and \(y \rightarrow b\)
  • iff the limit \(L\) is independent of the path followed by
  • the point \(x, y\) as \(x \rightarrow a\) and \(y \rightarrow b\)

\[ \lim_{(x, y) \rightarrow (a,b)} f(x, y) = L \]

Continuity

A function \(f(x, y)\) is said to be continuous at \((a, b)\) iff

\[ \lim_{(x, y) \rightarrow (a,b)} f(x, y) = f(a, b) \]

Homogeneous functions

• \(f(x, y)\) is homogeneous if it can be represented as

\[ f(x, y) = x^n f\left(\frac{x}{y}\right) \]

\(n\) is the called degree of the equation

Euler's theorem - for homogeneous equations

\[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y) \]

\[ x^2 \frac{\partial f}{\partial x^2} + 2xy \frac{\partial f}{\partial x \partial y} + y^2 \frac{\partial f}{\partial y^2} = n(n-1) f(x, y) \]

Expansion of function with multiple variables

Taylor Series

If \(f(x, y)\) and all its partial derivatives upto the \(n^{th}\) order are finite and continuous for all points \((x, y)\) where \(a \le x \le a + h\), \(b \le x \le b + h\) then

\[ f(a+h, b+k) = f(a, b) + \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)f + \frac{1}{2!} \left( h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y} \right)^2f + \dots \]

On putting \(a =0, b=0, h=x, k = y\)

\[ f(x, y) = f(0, 0) + \left( x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y} \right)f + \frac{1}{2!} \left( x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y} \right)^2f + \dots \]

Maxima and minima

• function \(f(x,y)\) have local maximum at point \((a, b)\)
  • iff \(f(a, b) > f(a+h, b+h)\) for some small values of \(h\)
  • similar for minimum
• find extreme value of a function
  • find \(f_x, f_y\)
  • solve \(f_x = 0, f_y = 0\) simultaneously
  • find \(f_{xx}, f_{xy}, f_{yy}\)
  • find \(D = f_{xx}f_{yy} - f_{xy}^2\) at each stationary point
    • if \(D>0\)
      • \(f_{xx} < 0\) local minimum
      • \(f_{xx} > 0\) local maximum
    • else if \(D<0\) no extreme values
    • else \(D=0\) test fails

Lagranage's Method of undetermined multipliers

• Used to find stationary point of function with multiple variables

Given function \(f(x, y, z)\), variable \(x, y, z\) are connected by the relation

\[ \phi(x, y, z) = 0 \]

Consider the lagrange function,

\[ F(x, y, z) = f(x, y, z) + \lambda \phi(x, y, z) \]

then for stationary point,

\(dF = 0\) $$ (f_{x} + \lambda \phi_{x}) dx + (f_{y} + \lambda \phi_{y}) dy + (f_{z} + \lambda \phi_{z}) dz = 0 $$ $$ \nabla F = 0 $$ $$ f_{x} + \lambda \phi_{x} = 0, f_{y} + \lambda \phi_{y} = 0, f_{z} + \lambda \phi_{z} = 0 $$

Jacobians

Functions \(u(x, y)\) and \(v(x, y)\) have Jacobian(functional determinant), which is

\[ J(u, v) = \frac{\partial(u, v)}{\partial(x, y)} = \begin{vmatrix} u_x & u_y \\ v_x & v_y \end{vmatrix} \]

Similarly for \(n\) functions \(u_1, u_2, \dots, u_n\), and \(n\) independent variables \(x_1, x_2, \dots, x_n\)

\[ \frac{\partial(u_1, u_2, \dots, u_n)}{\partial(x_1, x_2, \dots, x_n)} = \begin{vmatrix} (u_1)_{x_{1}} & (u_1)_{x_{2}} & \dots & (u_1)_{x_{n}} \\ (u_2)_{x_{1}} & (u_2)_{x_{2}} & \dots & (u_2)_{x_{n}} \\ \vdots & \vdots & \ddots & \vdots \\ (u_n)_{x_{1}} & (u_n)_{x_{2}} & \dots & (u_n)_{x_{n}} \\ \end{vmatrix} \]

Properties of Jacobians

• if \(u\) and \(v\) are functions of \(r\) and \(s\) and \(r\) and \(s\) are functions of \(x\) and \(y\) $$ \frac{\partial(u, v)}{\partial(x, y)} = \frac{\partial(u, v)}{\partial(r, s)} \frac{\partial(r, s)}{\partial(x, y)} $$
• If \(u_1, u_2, \dots, u_n\), and \(x_1, x_2, \dots, x_n\) are related as follows
  • \(u_1 = f_1(x_1)\)
  • \(u_2 = f_2(x_1, x_2)\)
  • \(\dots\)
  • \(u_n = f_n(x_1, x_2, \dots, x_n)\) then $$ \frac{\partial(u_1, u_2, \dots, u_n)}{\partial(x_1, x_2, \dots, x_n)} = \frac{u_1}{x_1} \frac{u_2}{x_2} \dots \frac{u_n}{x_n} $$
• For implicit functions - \(u_1, u_2, u_3\) are implicit functions of \(x_1, x_2, x_3\)
  • \(F_1(u_1, u_2, u_3, x_1, x_2, x_3) = 0\)
  • \(F_2(u_1, u_2, u_3, x_1, x_2, x_3) = 0\)
  • \(F_3(u_1, u_2, u_3, x_1, x_2, x_3) = 0\) $$ \frac{\partial(u_1, u_2, u_3)}{\partial (x_1, x_2, x_3)} = (-1)^3 \frac {\frac{\partial (F_1, F_2, F_3)}{\partial (x_1, x_2, x_3)}} {\frac{\partial (F_1, F_2, F_3)}{\partial (u_1, u_2, u_3)}} $$

Chain Rule

\[ \frac{\partial}{\partial x} f(g(x, y)) = \frac{d}{dg}f(g) \cdot \frac{\partial}{\partial x} g(x, y) \]