Multivariable Calculus
Partial derivatives #
Graph of a function with two variables: $$ (x,y,z) \in \mathbb{R}^3 \land z = f(x,y) \land (x, y) \in D $$ The set D is the domain of $f$
Partial derivative of $f$ w.r.t. $x$ at $(a, b)$:
$$ f_a(a,b) = g'(a) \text{ where } g(x)=f(x,b)=\lim_{h \to 0}\frac{f(a+h,b)-f(a,b)}{h} $$
Also works with implicit differentiation
Second partial derivatives:
$$ (f_x)_x = f_{xx} = f_{11} = \frac{\delta}{\delta x}(\frac{\delta f}{\delta x}) = \frac{\delta^2 f}{\delta x^2} $$
$$ (f_x)_y = f_{xy} = f_{12} = \frac{\delta}{\delta y}(\frac{\delta f}{\delta x}) = \frac{\delta^2 f}{\delta y\ \delta x} $$
$$ (f_y)_x = f_{yx} = f_{21} = \frac{\delta}{\delta x}(\frac{\delta f}{\delta y}) = \frac{\delta^2 f}{\delta x\ \delta y} $$
$$ (f_y)_y = f_{yy} = f_{22} = \frac{\delta}{\delta y}(\frac{\delta f}{\delta y}) = \frac{\delta^2 f}{\delta y^2} $$
Note that the order is reversed.
Clairaut's Theorem: Suppose $f$ is defined on a disk $D$ that contains the point $(a,b)$. In the functions $f_{xy}$ and $f_{yx}$ are both continuous on $D$, then
$$ f_{xy}(a,b) = f_{yx}(a,b) $$
Linear Approximations / Tangent plane #
Equation of the tangent plane to surface $z = f(x,y)$ at the point $P(x_0, y_0, z_0)$:
$$ z - z_0 = f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0) $$
Linear approximation / tangent plane approximation: $$ f(x,y) \approx f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y-b) $$
Tangent plane:
$$ \nabla f(x_0,y_0,z_0) \cdot \vec{r'}(t) = 0 $$
Plane equation: $$ f_x(x_0,y_0,z_0)(x - x_0) + f_y(x_0,y_0,z_0)(y - y_0) + f_z(x_0,y_0,z_0)(z - z_0) = 0 $$
Differentials #
Differential $dz$ / Total differential: $$ dz = f_x(x,y)dx + f_y(x,y)dy = \frac{\delta z}{\delta x}dx + \frac{\delta z}{\delta y}dy $$
The Chain Rule #
$z = f(x,y)$ where $x$ and $y$ are functions of $t$:
$$ \frac{dz}{dt} = \frac{\delta f}{\delta x}\frac{dx}{dt} + \frac{\delta f}{\delta y}\frac{dy}{dt} $$
$z = f(x,y)$ where $x = (s,y)$ and $y=h(s,y)$:
$$ \frac{\delta z}{\delta s} = \frac{\delta z}{\delta x}\frac{\delta x}{\delta s} + \frac{\delta z}{\delta y}\frac{\delta y}{\delta s} $$
$$ \frac{\delta z}{\delta t} = \frac{\delta z}{\delta x}\frac{\delta x}{\delta t} + \frac{\delta z}{\delta y}\frac{\delta y}{\delta t} $$
Can be extended to functions of any amount of variables.
Implicit derivative (Implicit Function Theorem):
$$ \frac{dy}{dx} = -\frac{\frac{\delta F}{\delta x}}{\frac{\delta F}{\delta y}} = -\frac{F_x}{f_y} $$
Directional derivative #
Theorem: If the partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$, then $f$ is differentiable at $(a,b)$
The directional derivative of $f$ at $(x_0, y_0)$ in the direction of the unit vector $\vec{u} = \langle a, b \rangle$:
$$ D_{\vec{u}} f(x_0,y_0) = \lim_{h \to 0}\frac{f(x_0+ha, y_0+hb) - f(x_0,y_0)}{h} $$
If $f$ is a differentiable function of $x$ and $y$, then $f$ has a directional derivative in the direction of any unit vector $\vec{u} = \langle a, b \rangle$: $$ D_{\vec{u}} f(x,y) = f_x(x,y)a + f_y(x,y)b $$
If the unit vector has an angle $\theta$ on the positive $x$-axis then $\vec{u} = \langle \cos\theta, \sin\theta \rangle$:
$$ D_{\vec{u}} f(x,y) = f_x(x,y)\cos\theta + f_y(x,y)\sin\theta $$
Gradients #
$$ \nabla f(x,y) = \langle f_x(x,y), f_y(x,y) \rangle = \frac{\delta f}{\delta x}\vec{i} + \frac{\delta f}{\delta y}\vec{j} $$
The directional derivative can be written as the dot product of two vectors (the gradient and unit vector):
$$ \begin{align*} D_{\vec{u}} f(x,y) &= f_x(x,y)a + f_y(x,y)b \\ &= \langle f_x(x,y), f_y(x,y) \rangle \cdot \langle a, b \rangle \\ &= \langle f_x(x,y), f_y(x,y) \rangle \cdot \vec{u} \\ &= \nabla f \cdot \vec{u} \end{align*} $$
Minimum and Maximum values #
Critical/stationary point $(a,b)$ if $f_x(a,b)=0$ and $f_y(a,b)=0$. Can be found by solving the system.
Local min/max #
- Local maximum at $(a,b)$ if $f(x,y) \leq f(a,b)$ when $f(x,y)$ is near $(a,b)$
- Local minimum at $(a,b)$ if $f(x,y) \geq f(a,b)$ when $f(x,y)$ is near $(a,b)$
The value $f(a,b)$ is called the local maximum/minimum value
Second derivatives test #
$$ D = D(a,b) = f_{xx}(a,b)f_{yy}(a,b) - [f_{xy}(a,b)]^2 $$
- Local minimum $f(a,b)$ if $D > 0$ and $f_{xx}(a,b) > 0$
- Local maximum $f(a,b)$ if $D > 0$ and $f_{xx}(a,b) < 0$
- Saddle point $(a,b)$ if $D < 0$
To remember $D$ you can write it as a determinant: $$ D = \begin{vmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{vmatrix} = f_{xx}f_{yy} - (f_{xy})^2 $$
Double Integrals #
Over rectangles:
If $f$ is continuous the rectangle $$ R = \{ (x,y) \mid a \leq x \leq b, c \leq x \leq d \} $$ then $$ \iint\limits_{R} f(x,y)~dA = \int_a^b\int_c^d f(x,y)~dy~dx = \int_c^d\int_a^b f(x,y)~dx~dy $$
Over general regions: the constant bounds must be on the outside
Where $f$ is continuous of region $D$
Type I (Function of $x$): $$ D = \{ (x,y) \mid a \leq x \leq b, g_1(x) \leq y \leq g_2(x) \} $$ $$ \iint\limits_{D} f(x,y)~dA = \int_a^b\int_{g_1(x)}^{g_2(x)} f(x,y)~dy~dx $$
Type II (Function of $y$): $$ D = \{ (x,y) \mid c \leq y \leq d, h_1(y) \leq x \leq h_2(y) \} $$ $$ \iint\limits_{D} f(x,y)~dA = \int_a^b\int_{h_1(y)}^{h_2(y)} f(x,y)~dx~dy $$
Multiple integrals (more dimensions): each integral eliminates a set of variables