Distance of two points in 3 dimensions
$|P_1P_2|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$
$\begin{align}|P_1P_2|^2 &=|P_1B|^2+|BP_2|^2\\&=|P_1A|^2+|AB|^2+|BP_2|^2\\&=(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2\end{align}$
Direction cosine
$\vec{a}=\langle a_1,a_2,a_3 \rangle$
The angles $\alpha,\beta,\gamma$ in $[0,\pi]$ that $\vec{a}$ makes with $x,y,z$ axes respectively are direction angles of $\vec{a}$.
$\Large\cos \alpha=\frac{\vec{a}\cdot \hat{i}}{|\vec{a}||\hat{i}|}=\frac{\langle a_1,a_2,a_3 \rangle \cdot \langle 1,0,0 \rangle}{|\langle a_1,a_2,a_3 \rangle||\langle 1,0,0 \rangle|}=\frac{a_1}{|\vec{a}|}$
Similarly, $\large \cos \beta=\frac{a_2}{|\vec{a}|}$ and $\large \cos \gamma=\frac{a_3}{|\vec{a}|}$.
Thus, $\vec{a}=|\vec{a}|\langle \cos\alpha,\cos\beta,\cos\gamma \rangle$
Note also that $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=\frac{1}{|\vec{a}|^2}(a_1^2+a_2^2+a_3^2)=1$
Dot product in component form
Cosine Law: $\begin{align} |\vec{a}-\vec{b}|^2&=|\vec{a}|^2+|\vec{b}|^2-2|\vec{a}||\vec{b}|\cos \theta\\ &=|\vec{a}|^2+|\vec{b}|^2-2\vec{a}\cdot\vec{b}\end{align}$
$\begin{align} \vec{a}\cdot \vec{b}&=|\vec{a}||\vec{b}|\cos \theta\\
&=\frac{1}{2}(|\vec{a}|^2+|\vec{b}|^2-|\vec{a}-\vec{b}|^2)\\&=\frac{1}{2}[a_1^2+a_2^2+a_3^2+b_1^2+b_2^2+b_3^2-(a_1-b_1)^2-(a_2-b_2)^2-(a_3-b_3)^2]\\&=a_1b_1+a_2b_2+a_3b_3 \end{align}$
Projections
scalar projection of $\vec{b}$ onto $\vec{a}$:
$\large \text{comp}_\vec{a} \vec{b}=|\vec{b}|\cos \theta=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}|}=\frac{\vec{a}}{|\vec{a}|}\cdot \vec{b}$
vector projection of $\vec{b}$ onto $\vec{a}$:
$\large \text{proj}_\vec{a} \vec{b}=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}|} \cdot \frac{\vec{a}}{|\vec{a}|}=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}|^2} \vec{a}$
Scalar triple product
volume of a parallelepiped $V=Ah=|\vec{b}\times\vec{c}||\vec{a}|\cos \theta=|\vec{a}\cdot (\vec{b} \times \vec{c})|$
The base parallelogram can be determined by [$\vec{b}$ and $\vec{c}$] or [$\vec{a}$ and $\vec{b}$].
Also, because of the commutative property of dot products, we have $\vec{a}\cdot (\vec{b} \times \vec{c})=(\vec{a} \times \vec{b}) \cdot \vec{c}$.
symmetric equation, vector equation, scalar equation of the plane through a point with normal vector, linear equation
skew lines
More to know [later]
Orthogonal projection
$\large \text{ortho}_\vec{a} \vec{b}=\vec{b}-\text{proj}_\vec{a} \vec{b}$
Parallelogram Law
$|\vec{a}+\vec{b}|^2+|\vec{a}-\vec{b}|^2=2|\vec{a}|^2+2|\vec{b}|^2$
geometric meaning
Vector and chemistry
Bond angle 109.5
Vector triple product
$\vec{a}\times\vec{b}\times\vec{c}=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}$
$\Rightarrow \vec{a}\times(\vec{b}\times\vec{c})+\vec{b}\times(\vec{c}\times\vec{a})+\vec{c}\times(\vec{a}\times\vec{b})=\vec{0}$
$(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})=\begin{vmatrix}\vec{a}\cdot\vec{c} & \vec{b}\cdot\vec{c}\\ \vec{a}\cdot\vec{d} & \vec{b}\cdot\vec{d} \end{vmatrix}$
crystallography
Useful:
http://www.math.jhu.edu/~hhaosu/Teaching/0607FCalcIII/
http://www.math.jhu.edu/~hhaosu/Teaching/0607FCalcIII/06FMA261AHW02Sol.pdf
http://www.leadinglesson.com/problem-on-orthogonal-matrices
Friday, 17 April 2015
Friday, 10 April 2015
Parametrizations
Line segments
Given two points $a,b$ in $\Bbb C$,the line segment $ab$ has parametric equation $$\color{blue}{z(t)=(1-t)a+tb},\quad t\in [0,1].$$
Circles
We know the rectangular equation for a circle with center at the origin: $$x^2+y^2=r^2.$$ The equality also holds when we substitute $$\color{blue}{x=r\cos t,\quad y=r\sin t}.$$ What about parametrizing a circle with center $(a,b)$? We have $$(x-a)^2+(y-b)^2=r^2.$$ We can put $$\color{blue}{x=r\cos t+a,\quad y=r\sin t+b}.$$ In the complex plane, we have $$z(t)=re^{it},\quad t\in [0,2\pi]$$ for circles with center at the origin and $$\color{blue}{z(t)=z_0+re^{it}},\quad t\in [0,2\pi]$$ for circles with center $z_0$.
Ellipses
A ellipse is the set of points $P$ such that the sum of whose distances from two fixed points (the foci $F_1$ and $F_2$) separated by a distance $2c$ is a given positive constant $2a$. It is given by $$\{P:|d(P,F_1)+d(P,F_2)|=2a\}.$$ When $F_1=(-c,0)$ and $F_2=(c,0)$, we have $$\bigg\{(x,y) \;\bigg|\; \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\bigg\}, \quad b=\sqrt{c^2-a^2}$$ with parametrization $$\color{blue}{x=a\cos t, \quad y=b\sin t},\quad t\in [0,2\pi]$$
Hyperbolas
A hyperbola is the set of points $P$ in a plane that the difference of whose distances from two fixed points (the foci $F_1$ and $F_2$) separated by a distance $2c$ is a given positive constant $2a$. It is given by $$\{P:|d(P,F_1)-d(P,F_2)|=2a\}.$$ When $F_1=(-c,0)$ and $F_2=(c,0)$, we have $$\bigg\{(x,y)\;\bigg| \;\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\bigg\}, \quad b=\sqrt{c^2-a^2}.$$ Recall that $\cosh^2 t-\sinh^2 t=1$. The right branch is parametrized by $$\color{blue}{x=a\cosh t,\quad y=b\sinh t},\quad t\in (-\infty,\infty).$$ The left branch is parametrized by $$\color{blue}{x=-a\cosh t,\quad y=b\sinh t},\quad t\in (-\infty,\infty).$$
More to know
Lemniscates
A lemniscate is the set of points $P$ in a plane that the product of whose distances from two fixed points (the foci $F_1$ and $F_2$) a distance $2c$ away is the constant $c^2$.
Minimal surfaces of revolution
Let $\gamma:I\to \Bbb R^2$ be a parametrized curve of the form $\gamma(t)=(t,y(t))$ for some smooth function $y:I\to \Bbb R$ with $y(t)>0$ for all $t\in I$. The surface obtained by rotating $\gamma $ about the $x$-axis is given by $$\phi(t,\theta)=(t,y(t)\cos\theta,y(t)\sin\theta).$$ Reference
Parametric curves
Given two points $a,b$ in $\Bbb C$,the line segment $ab$ has parametric equation $$\color{blue}{z(t)=(1-t)a+tb},\quad t\in [0,1].$$
Circles
We know the rectangular equation for a circle with center at the origin: $$x^2+y^2=r^2.$$ The equality also holds when we substitute $$\color{blue}{x=r\cos t,\quad y=r\sin t}.$$ What about parametrizing a circle with center $(a,b)$? We have $$(x-a)^2+(y-b)^2=r^2.$$ We can put $$\color{blue}{x=r\cos t+a,\quad y=r\sin t+b}.$$ In the complex plane, we have $$z(t)=re^{it},\quad t\in [0,2\pi]$$ for circles with center at the origin and $$\color{blue}{z(t)=z_0+re^{it}},\quad t\in [0,2\pi]$$ for circles with center $z_0$.
Ellipses
A ellipse is the set of points $P$ such that the sum of whose distances from two fixed points (the foci $F_1$ and $F_2$) separated by a distance $2c$ is a given positive constant $2a$. It is given by $$\{P:|d(P,F_1)+d(P,F_2)|=2a\}.$$ When $F_1=(-c,0)$ and $F_2=(c,0)$, we have $$\bigg\{(x,y) \;\bigg|\; \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\bigg\}, \quad b=\sqrt{c^2-a^2}$$ with parametrization $$\color{blue}{x=a\cos t, \quad y=b\sin t},\quad t\in [0,2\pi]$$
Hyperbolas
A hyperbola is the set of points $P$ in a plane that the difference of whose distances from two fixed points (the foci $F_1$ and $F_2$) separated by a distance $2c$ is a given positive constant $2a$. It is given by $$\{P:|d(P,F_1)-d(P,F_2)|=2a\}.$$ When $F_1=(-c,0)$ and $F_2=(c,0)$, we have $$\bigg\{(x,y)\;\bigg| \;\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\bigg\}, \quad b=\sqrt{c^2-a^2}.$$ Recall that $\cosh^2 t-\sinh^2 t=1$. The right branch is parametrized by $$\color{blue}{x=a\cosh t,\quad y=b\sinh t},\quad t\in (-\infty,\infty).$$ The left branch is parametrized by $$\color{blue}{x=-a\cosh t,\quad y=b\sinh t},\quad t\in (-\infty,\infty).$$
More to know
Lemniscates
A lemniscate is the set of points $P$ in a plane that the product of whose distances from two fixed points (the foci $F_1$ and $F_2$) a distance $2c$ away is the constant $c^2$.
Minimal surfaces of revolution
Let $\gamma:I\to \Bbb R^2$ be a parametrized curve of the form $\gamma(t)=(t,y(t))$ for some smooth function $y:I\to \Bbb R$ with $y(t)>0$ for all $t\in I$. The surface obtained by rotating $\gamma $ about the $x$-axis is given by $$\phi(t,\theta)=(t,y(t)\cos\theta,y(t)\sin\theta).$$ Reference
Parametric curves
Thursday, 2 April 2015
Naming quadric surfaces
Non-degenerate real quadric surfaces
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-z=0\\
z=\frac{x^2}{a^2}+\frac{y^2}{b^2}$
Consider the horizontal cross section (level curve) of z by setting $z=c$.
We have $\frac{x^2}{a^2}+\frac{y^2}{b^2}=c$, which is the equation for an ellipse.
Now consider the vertical cross section of z at $x=c$.
We get $z=\frac{c^2}{a^2}+\frac{y^2}{b^2}$ -- a constant plus $\frac{y^2}{b^2}$, which is the equation of a parabola. Similarly, the vertical cross section of z at $y=c$ is also a parabola.
We know that the figure is a elliptical paraboloid.
When we change b (denominator) to a, we have a circular paraboloid.
$\frac{x^2}{a^2}-\frac{y^2}{b^2}-z=0\\
z=\frac{x^2}{a^2}-\frac{y^2}{b^2}$
Horizontal cross section: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=c\:\:[z=c]\Rightarrow$ hyperbola
Vertical cross section: $\frac{x^2}{a^2}- \text{constant}$ or $\text{constant} - \frac{y^2}{b^2} \Rightarrow$ parabola
Surface: Hyperbolic paraboloid
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$
Horizontal cross section: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{k^2}{c^2}+1\:\:[z=k] \Rightarrow$ ellipse
Vertical cross section: $\frac{x^2}{a^2}-\frac{z^2}{c^2}=1-\frac{m^2}{b^2}\:\:[y=m]$ or $\frac{y^2}{b^2}-\frac{z^2}{c^2}=1-\frac{n^2}{a^2}\:\:[x=n] \Rightarrow$ hyperbola
Surface: Elliptical hyperboloid of one sheet
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1$
Horizontal cross section: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{k^2}{c^2}-1\:\:[z=k] \Rightarrow$ generally ellipse
There are no horizontal cross sections when z is between -1 and 1. For simplicity, we can look at the equation $-x^2-y^2+z^2=1$. Suppose $z=0$. Then the left hand side is negative, but the right hand side is positive. There's no way to fix this, so the cross section simply doesn't exist!
Vertical cross section: $\frac{z^2}{c^2}-\frac{x^2}{a^2}=1+\frac{m^2}{b^2}\:\:[y=m]$ or $\frac{z^2}{c^2}-\frac{y^2}{b^2}=1+\frac{n^2}{a^2}\:\:[x=n] \Rightarrow$ hyperbola
Surface: Elliptical hyperboloid of two sheets
Here's a hint about telling the two kinds of hyperboloids apart: look at the cross sections x=0, y=0, and z=0 (respectively but not simultaneously). If they exist, then it's a hyperboloid of one sheet. If you end up with something negative equal to something positive, it's a hyperboloid of two sheets.
Degenerate quadric surfaces
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0$
Horizontal cross section: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{k^2}{c^2}\:\:[z=k] \Rightarrow$ ellipse
Vertical cross section:
(1) $\frac{z^2}{c^2}-\frac{x^2}{a^2}=\frac{m^2}{b^2}\:\:[y=m]$ or $\frac{z^2}{c^2}-\frac{y^2}{b^2}=\frac{n^2}{a^2}\:\:[x=n] \Rightarrow$ hyperbola
(2) $\frac{x^2}{a^2}-\frac{z^2}{c^2}=0\:\:[y=0]$ or $\frac{y^2}{b^2}-\frac{z^2}{c^2}=0\:\:[x=0]$ $\iff (\frac{x}{a}-\frac{z}{c})(\frac{x}{a}+\frac{z}{c})=0$ or $(\frac{y}{b}-\frac{z}{c})(\frac{y}{b}+\frac{z}{c})=0 \Rightarrow$ pair of straight lines
Surface: Elliptic Cone
$\frac{x^2}{a^2}+\frac{y^2}{a^2}-\frac{z^2}{b^2}=0$ -- Special case of cone (Circular cone)
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ -- Elliptic cylinder
$\frac{x^2}{a^2}+\frac{y^2}{a^2}=1$ -- Circular cylinder
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ -- Hyperbolic cylinder
Useful website:
Quadric surfaces
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-z=0\\
z=\frac{x^2}{a^2}+\frac{y^2}{b^2}$
Consider the horizontal cross section (level curve) of z by setting $z=c$.
We have $\frac{x^2}{a^2}+\frac{y^2}{b^2}=c$, which is the equation for an ellipse.
Now consider the vertical cross section of z at $x=c$.
We get $z=\frac{c^2}{a^2}+\frac{y^2}{b^2}$ -- a constant plus $\frac{y^2}{b^2}$, which is the equation of a parabola. Similarly, the vertical cross section of z at $y=c$ is also a parabola.
We know that the figure is a elliptical paraboloid.
When we change b (denominator) to a, we have a circular paraboloid.
$\frac{x^2}{a^2}-\frac{y^2}{b^2}-z=0\\
z=\frac{x^2}{a^2}-\frac{y^2}{b^2}$
Horizontal cross section: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=c\:\:[z=c]\Rightarrow$ hyperbola
Vertical cross section: $\frac{x^2}{a^2}- \text{constant}$ or $\text{constant} - \frac{y^2}{b^2} \Rightarrow$ parabola
Surface: Hyperbolic paraboloid
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$
Horizontal cross section: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{k^2}{c^2}+1\:\:[z=k] \Rightarrow$ ellipse
Vertical cross section: $\frac{x^2}{a^2}-\frac{z^2}{c^2}=1-\frac{m^2}{b^2}\:\:[y=m]$ or $\frac{y^2}{b^2}-\frac{z^2}{c^2}=1-\frac{n^2}{a^2}\:\:[x=n] \Rightarrow$ hyperbola
Surface: Elliptical hyperboloid of one sheet
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1$
Horizontal cross section: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{k^2}{c^2}-1\:\:[z=k] \Rightarrow$ generally ellipse
There are no horizontal cross sections when z is between -1 and 1. For simplicity, we can look at the equation $-x^2-y^2+z^2=1$. Suppose $z=0$. Then the left hand side is negative, but the right hand side is positive. There's no way to fix this, so the cross section simply doesn't exist!
Vertical cross section: $\frac{z^2}{c^2}-\frac{x^2}{a^2}=1+\frac{m^2}{b^2}\:\:[y=m]$ or $\frac{z^2}{c^2}-\frac{y^2}{b^2}=1+\frac{n^2}{a^2}\:\:[x=n] \Rightarrow$ hyperbola
Surface: Elliptical hyperboloid of two sheets
Here's a hint about telling the two kinds of hyperboloids apart: look at the cross sections x=0, y=0, and z=0 (respectively but not simultaneously). If they exist, then it's a hyperboloid of one sheet. If you end up with something negative equal to something positive, it's a hyperboloid of two sheets.
Degenerate quadric surfaces
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0$
Horizontal cross section: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{k^2}{c^2}\:\:[z=k] \Rightarrow$ ellipse
Vertical cross section:
(1) $\frac{z^2}{c^2}-\frac{x^2}{a^2}=\frac{m^2}{b^2}\:\:[y=m]$ or $\frac{z^2}{c^2}-\frac{y^2}{b^2}=\frac{n^2}{a^2}\:\:[x=n] \Rightarrow$ hyperbola
(2) $\frac{x^2}{a^2}-\frac{z^2}{c^2}=0\:\:[y=0]$ or $\frac{y^2}{b^2}-\frac{z^2}{c^2}=0\:\:[x=0]$ $\iff (\frac{x}{a}-\frac{z}{c})(\frac{x}{a}+\frac{z}{c})=0$ or $(\frac{y}{b}-\frac{z}{c})(\frac{y}{b}+\frac{z}{c})=0 \Rightarrow$ pair of straight lines
Surface: Elliptic Cone
$\frac{x^2}{a^2}+\frac{y^2}{a^2}-\frac{z^2}{b^2}=0$ -- Special case of cone (Circular cone)
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ -- Elliptic cylinder
$\frac{x^2}{a^2}+\frac{y^2}{a^2}=1$ -- Circular cylinder
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ -- Hyperbolic cylinder
Useful website:
Quadric surfaces
Wednesday, 1 April 2015
Directional derivatives
The partial derivatives $f_x(x_0,y_0)$ and $f_y(x_0,y_0)$ are the rate of changes of $z=f(x,y)$ at $(x_0,y_0)$ in the positive x- and y- directions. Rate of change in other directions are given by directional derivatives. In fact, the partial derivatives of f with respect to x and y are just special cases of the directional derivative.
To find the directional derivatives of $z=f(x,y)$ at $(x_0,y_0)$ in the direction of the unit vector $\vec{u}=\langle u_1, u_2 \rangle$ in the xy-plane, we introduce an s-axis with its origin at $(x_0,y_0)$, with its positive direction in the direction of $\vec{u}$. Then the point at s on the s-axis has xy-coordinates $x=x_0+su_1, y=y_0+su_2$, and the value of $z = f(x, y)$ at the point s on the s-axis is $F(s)=f(x_0+su_1,y_0+su_2)$. We call $z=F(s)$ the cross section through $(x_0,y_0)$ of $z=f(x,y)$ in the direction of $\vec{u}$.
If the origin is not $(0,0)$, we introduce a second vertical z-axis (in blue) with its origin at the point $(x_0, y_0,0)$ (the origin on the s-axis). Then the graph of $z=F(s)$ is the intersection of the surface $z=f(x,y)$ with the sz-plane. The directional derivative of $z=f(x,y)$ is the slope of the tangent line to this curve in the positive s-direction at s = 0, which is at the point $(x_0, y_0, f(x_0, y_0))$.
The directional derivative [denoted $D_{\vec{u}}f(x_0,y_0)$] of $z=f(x,y)$ at $(x_0,y_0)$ in the direction of the unit vector $\vec{u}=\langle u_1,u_2 \rangle$ is the derivative of the cross section function (1) at $s=0$:
$D_{\vec{u}}f(x_0,y_0)=\frac{d}{ds}f(x_0+su_1,y_0+su_2)|_{s=0}$
For any unit vector $u=\langle u_1,u_2 \rangle$, the (directional) derivative of $z=f(x, y)$ at $(x_0, y_0)$ in the direction of $\vec{u}$ is $D_{\vec{u}}f(x_0,y_0)=f_x(x_0,y_0)u_1+f_y(x_0,y_0)u_2$
Proof:
$F'(s)\\=\frac{d}{ds}[f(x_0+su_1,y_0+su_2)]\\=f_x(x_0+su_1,y_0+su_2)\frac{d}{ds}(x_0+su_1)+f_y(x_0+su_1,y_0+su_2)\frac{d}{ds}(y_0+su_2)\\=f_x(x_0+su_1,y_0+su_2)u_1+f_y(x_0+su_1,y_0+su_2)u_2\\=f_x(x_0,y_0)u_1+f_y(x_0,y_0)u_2 \:\:\text{[put s=0]}$
To find the directional derivatives of $z=f(x,y)$ at $(x_0,y_0)$ in the direction of the unit vector $\vec{u}=\langle u_1, u_2 \rangle$ in the xy-plane, we introduce an s-axis with its origin at $(x_0,y_0)$, with its positive direction in the direction of $\vec{u}$. Then the point at s on the s-axis has xy-coordinates $x=x_0+su_1, y=y_0+su_2$, and the value of $z = f(x, y)$ at the point s on the s-axis is $F(s)=f(x_0+su_1,y_0+su_2)$. We call $z=F(s)$ the cross section through $(x_0,y_0)$ of $z=f(x,y)$ in the direction of $\vec{u}$.
If the origin is not $(0,0)$, we introduce a second vertical z-axis (in blue) with its origin at the point $(x_0, y_0,0)$ (the origin on the s-axis). Then the graph of $z=F(s)$ is the intersection of the surface $z=f(x,y)$ with the sz-plane. The directional derivative of $z=f(x,y)$ is the slope of the tangent line to this curve in the positive s-direction at s = 0, which is at the point $(x_0, y_0, f(x_0, y_0))$.
The directional derivative [denoted $D_{\vec{u}}f(x_0,y_0)$] of $z=f(x,y)$ at $(x_0,y_0)$ in the direction of the unit vector $\vec{u}=\langle u_1,u_2 \rangle$ is the derivative of the cross section function (1) at $s=0$:
$D_{\vec{u}}f(x_0,y_0)=\frac{d}{ds}f(x_0+su_1,y_0+su_2)|_{s=0}$
For any unit vector $u=\langle u_1,u_2 \rangle$, the (directional) derivative of $z=f(x, y)$ at $(x_0, y_0)$ in the direction of $\vec{u}$ is $D_{\vec{u}}f(x_0,y_0)=f_x(x_0,y_0)u_1+f_y(x_0,y_0)u_2$
Proof:
$F'(s)\\=\frac{d}{ds}[f(x_0+su_1,y_0+su_2)]\\=f_x(x_0+su_1,y_0+su_2)\frac{d}{ds}(x_0+su_1)+f_y(x_0+su_1,y_0+su_2)\frac{d}{ds}(y_0+su_2)\\=f_x(x_0+su_1,y_0+su_2)u_1+f_y(x_0+su_1,y_0+su_2)u_2\\=f_x(x_0,y_0)u_1+f_y(x_0,y_0)u_2 \:\:\text{[put s=0]}$
Friday, 6 March 2015
Inclusion–exclusion principle
When we number the regions of different sets, we can easily see that
$\underbrace{n(A\cup B \cup C)}_{1234567}\\
=\underbrace{n(A)}_{1347}+\underbrace{n(B)}_{1246}+\underbrace{n(C)}_{1235}-\underbrace{n(A\cap B)}_{14}-\underbrace{n(A\cap C)}_{13}-\underbrace{n(B\cap C)}_{12}+\underbrace{(A\cap B\cap C)}_{1}$
The elements in each of regions 1 through 7 are counted once on both the LHS and RHS.
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