Partial Derivative

Definition:

• Differentiation respect to multiple variables

Functions of several variables: $z=f(x,y)$

• $\{f(x,y)|(x,y)\in D\}$
• Graph:
  • Draw $f(x,y)=6-3x-2y$
    • Then let $z=f(x,y)$
    • Use quadric surface to draw $\to 3x+2y+z=6$
• Level curve:
  • Contour map consists of many level curves for $k=a,b,c,..$ would be the curve that represents all the points in the function where $f(x,y)=a\text{ and }f(x,y)=b\text{ and }f(x,y))=c$
• Functions of three or more variables:
  • $D=f(x,y,z)$
  • 4-dimensional shape

Limits and continuity:

• Find limit of $\underset{(x,y)\to (a,b)}{\lim }f(x,y)$:
  • Approach from left side, $L_1=\underset{x\to 0}{\lim }f(x,0)$
  • Approach from right side, $L_2=\underset{y\to 0}{\lim }f(0,y)$
  • Approach from $$\begin{gathered} y-b=m(x-a), \\ {L}_3=\underset{x\to 0}{\lim }f(x,m(x-a)+b) \end{gathered}$$
  • Approach from $y=x^2,L_3=\underset{x\to 0}{\lim }f(x,x^2)$
  • Use different techniques to find limit
  • Limit exists all limits are the same
    • Be careful of the ridge
• $\underset{(x,y)\to (a,b)}{\lim }f(x,y)=L$
• Continuity:
  • A polynomial function of two variables sum of terms in form of $c{x}^m{y}^n$
    • is a continuous function
  • A rational function ratio of 2 polynomial
    • is a continuous function if denominator is not 0
  • Continuous function’s limit can be found by direct substitution
  • If $f(x,y)$ and $g(t)$ are both continuous then $h(x,y)=g(f(x,y))$ is also a continuous function

Partial derivatives:

• $D_{x}f=f_x(a,b)=g^{\prime }(a)$
  • Partial derivative of $f$ with respect to $x$ at $(a,b)$, where $g(x)=f(x,b)$
    • Represent the tangent line where $y=b$
  • Regard $y$ as a constant, then function of $x,y$ becomes function of $x$
  • Same for $y$
    • $f_y(a,b)=h^{\prime }(a)$ where $h(y)=f(a,y)$
  • Differentiate implicitly but for 3 variables
• $$\begin{gathered} g^{\prime }(a)=f_x(x,y)=\underset{h\to 0}{\lim }\frac{g(x+h)-g(x)}{h}= \\ \underset{h\to 0}{\lim }\frac{f(x+h,y)-f(x,y)}{h} \end{gathered}$$
• Functions with more than 2 variables:
  • Treat the other 2 variables as constants
    • ex: $f(x,y,z)=e^{xy}\ln z⟹f_x=y{e}^{xy}\ln z$
• Higher derivatives:
  • $(f_x{)}_x=f_{xx}=f_{11}=\frac{\partial }{\partial x}(\frac{\partial f}{\partial x})=\frac{{\partial }^{2}f}{\partial x^2}=\frac{{\partial }^{2}z}{\partial x^2}$
    • Differentiate $f_x$ in term of $x$ again
  • $(f_x{)}_y=f_{xy}=f_{12}=\frac{\partial }{\partial y}(\frac{\partial f}{\partial x})=\frac{{\partial }^{2}f}{\partial y\partial x}=\frac{{\partial }^{2}z}{\partial y\partial x}$
    • Differentiate $f_x$ in term of $y$ again
  • $f_{x}y(a,b)=f_{y}x(a,b)$
• Partial differential equation:
  • Express certain physical laws
    • like Laplace equation $(u_{xx}+u_{yy}=0)$
    • and wave equation $(u_{tt}=a^2{u}_{xx})$

Tangent planes and linear approximations:

• Tangent planes:
  • Let $C_1$ and $C_2$ be curves obtained by intersecting vertical lines $y=y_0$ and $x=x_0$
  • Let $T_1$ and $T_2$ be tangent line to $C_1$ and $C_2$
  • $z=f(x,y)$ then the tangent plane at $P(x_0,y_0,z_0)$ is $z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$
• Linear approximations:
  • Linearization: $L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$
  • Linearization: $L(x,y,z)=f(a,b,c)+f_{x_1}(x-a)+f_{y_1}(y-b)+f_{z_1}(z-c)$
  • If the partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$ then $f$ is continuous at $f(a,b)$
• Differentials:
  • $dz=f_x(x,y)dx+f_y(x,y)dy=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$
• Functions of three or more variables:
  • $dw=\frac{\partial w}{\partial x}dx+\frac{\partial w}{\partial y}dy+\frac{\partial w}{\partial z}dz$

The chain rule:

• Case 1 (simple parametric):
  • $z=f(x,y)$ is a differentiable function and $x=g(t),y=h(t)$ are both differentiable $⟹\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}$
• Case 2 (two variable parametric):
  • $z=f(x,y)$ is a differentiable function and $x=g(s,t),y=h(s,t)$ are both differentiable
    • $⟹\frac{\partial z}{\partial s}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial s}$
    • $⟹\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$
  • $s,t$ are independent variables; $x,y$ are intermediate variables; $z$ is dependent variables
• General version:
  • $\frac{\partial w}{\partial u}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial u}+\frac{\partial w}{\partial z}\frac{\partial z}{\partial u}+\frac{\partial w}{\partial t}\frac{\partial t}{\partial u}+...$
  • $\frac{\partial w}{\partial v}=\frac{\partial w}{\partial x}\frac{\partial x}{\partial v}+\frac{\partial w}{\partial y}\frac{\partial y}{\partial v}+\frac{\partial w}{\partial z}\frac{\partial z}{\partial v}+\frac{\partial w}{\partial t}\frac{\partial t}{\partial v}+...$
  • same for more independent variables
• Implicit differentiation $F(x,y)=F(x,fx)=0$:
  • Case 1:
    • Suppose that $y$ is given implicitly as a function by an equation $y=f(x)$ by equation of the form $F(x,y)=F(x,f(x))=0$
      • $⟹\frac{dy}{dx}=-\frac{F_x}{F_y}$
  • Case 2:
    • Suppose that $z$ is given implicitly as a function by an equation $z=f(x,y)$ by equation of the form $F(x,y,z)=F(x,y,f(x,y))=0$
      • $⟹\frac{\partial z}{\partial x}=-\frac{\partial F}{\partial x}/\frac{\partial F}{\partial z}=-\frac{F_x}{F_z}$
      • $⟹\frac{\partial z}{\partial y}=-\frac{\partial F}{\partial y}/\frac{\partial F}{\partial z}=-\frac{F_y}{F_z}$

Directional derivatives & the gradient vector:

• $D_{u}f(x_0,y_0)=\underset{h\to 0}{\lim }\frac{f(x_0+ha,y_0+hb)-f(x_0,y_0)}{h}=\nabla f(\mathbf{x}).\hat{u}$
• Directional derivative:
  • $D_{u}f(x_0,y_0)$ denote directional derivative $f$ at $(x_0,y_0)$ in the direction $u$
  • $f_x$ and $f_y$ are special cases of directional derivative:
    • $$\begin{gathered} D_{u}f(x,y)=f_x(x,y).a+f_y(x,y).b \\ =f_x(x,y)\cos \theta +f_y(x,y)\sin \theta \end{gathered}$$
      • bcz $a=\mathbf{u}\cos \theta$
      • where $\theta$ is the angle $\mathbf{u}$ makes with $x$-axis
• Gradient vector : $\nabla f(x,y)$
  • Gives the direction of fastest increase of $f$
  • also the directional of the line orthogonal to the level surface $S$ of $f$ through $P$
  • also the perpendicular line to the level curve $f(x,y)=k$
  • By dot product: $$\begin{gathered} D_{u}f(x,y)=f_x(x,y).a+f_y(x,y).b \\ =\underset{\nabla f(x,y)}{\underbrace{⟨f_x(x,y)+f_y(x,y)⟩}}\cdot \mathbf{u} \end{gathered}$$
  • Gradient vector: $\nabla f(x,y)=⟨f_x,f_y⟩=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}$
• Functions of three variables:
  • Directional vector
    • $$\begin{gathered} D_{u}f(x,y,z)=f_x(x,y,z).a+f_y(x,y,z).b+f_z(x,y,z).c \\ =\nabla f(x,y,z)\cdot \mathbf{u} \end{gathered}$$
  • Gradient vector represents the normal line pokes perpendicular through the surface
    • $\nabla f(x,y,z)=⟨f_x,f_y,f_x⟩=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}+\frac{\partial k}{\partial y}\mathbf{k}$
• Maximizing the directional derivative:
  • The maximum value of the directional derivative is $D_{\mathbf{u}}f(x,...)\text{ is }|\nabla f(x,...)|$
    • occurs when $\mathbf{u}$ is the unit vector of gradient vector $\nabla f(x,...)$
• Tangent planes to level surfaces:
  • Tangent plane:
    • Let $x,y,z$ be functions of $$\begin{gathered} t \\ \to F(x(t),y(t),z(t))=k \\ \to \frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{\partial k}{\partial y}\frac{dz}{dt}=\nabla F\cdot {\mathbf{r}}^{\prime }(t)=0 \\ \to F_{x0}(x-x_0)+F_{y0}(y-y_0)+F_{z0}(z-z_0)=0 \end{gathered}$$ is the equation of tangent plane to the level surface $F(x,y,z)=k$ at $P(x_0,y_0,z_0)$
  • Symmetric equations of normal line to S at P:
    • $\frac{x-x_0}{F_x}=\frac{y-y_0}{F_y}=\frac{z-z_0}{F_z}$

Maximum and minimum values:

• Absolute minimum/maximum value:

  • $\nabla f(a,b)=f_x(a,b)=f_y(a,b)=0$ then $(a,b)$ is critical point

• Second derivatives test:

  • $D=\left|\begin{array}{cc}{f}_{xx}& f_{xy}\\ f_{yx}& f_{yy}\end{array}\right|=f_{xx}{f}_{yy}-(f_{xy}{)}^2$

  1. $D>0$ and $f_{xx}>0$ then $f(a,b)$ is a local minimum
  2. $D>0$ and $f_{xx}<0$ then $f(a,b)$ is a local maximum
  3. $D<0$ then $f(a,b)$ is not a local maximum or minimum, it is a saddle point
     • $f$ is maximum in $x$ direction but minimum in $y$ direction or vice versa

• Closed set vs bounded set

• Extreme value theorem for functions of two variables:

  • If $f$ is continuous on a closed, bounded set $D$ in ${\mathbb{R}}^2$, then $f$ attains an absolute max value $f(x_1,y_1)$ and an absolute min value $f(x_2,y_2)$ at some points in $D$

• To find absolute max and min for bounded set:

  1. Find the values of $f$ at the critical points of $f$ in $D$
  2. Find the extreme values of $f$ on the boundary
     • when $x=x_{min}/x_{max}$ and $y=y_{min}/y_{max}$
  3. Compare the values

Lagrange Multipliers