Analytic Geometry Of $\mathbb{R}^3$
Vector Products
The Dot Product
The dot product is give by
where $\theta$ is the angle between $A$ and $B$.
The Cross Product
The corss product $A\times B$ is the vector that’s perpendicular by right-hand rule to the plane containing $A$ and $B$ and whose magnitude is $|A||B|\sin\theta$.
$|A\times B|$ is the area of the parallelogram formed by $A$ and $B$.
Also, we have
where $i,j,k$ are the standard coordinate unit basis.
The Triple Scalar Product
The triple scalar product is $(A\times B)\cdot C$. $|(A\times B)\cdot C|$ is the volume of parallelepiped formed by $A$, $B$ and $C$. Also,
Geometry
Lines
The distance $d$ from a point $({ x }_{ 0 },{ y }_{ 0 })$ to the line $Ax+By=C$ is given by
The nearest point to the origin in the line $Ax+By=C$ is given by and the corresponding distance is
If we want to find the nearest point in the line to a point $(x_0,y_0)$, we can first transform to a new coordinate system by setting $x’=x-x_0$ and $y’=y-y_0$. The line becomes $Ax’+By’=C-Ax_0-By_0$. Then the nearest point in the new coordinate system is given by
By transforming it back to the original system, we have
Planes
A convinient way to represent the equation for a plane is by $\mathbf{n}\cdot (\mathbf{x} − \mathbf{p}) = 0$, where $\mathbf{p}$ is the vector to a point in the plane, and $\mathbf{n}$ is a normal vector to the plane.
The nearest point to the origin in the plane $Ax+By+Cz=D$ is given by
and the distance $d$ from the origin to this plane is given by
If we want to find the nearest point in the plane to a point $(x_0,y_0)$, we can apply similar method as for a line to obtain similar results.
The Intercept of Two Planes
Two planes in $\mathbb{R}^3$ are parallel, coincide, or intersect in a line. To calculate the intercept line, we need to eliminate $x$, $y$ or $z$ to get relationship between each two of them.
The Tangent Plane To A Surface
The equation of the tangent plane to the surface $z=f(x,y)$ at $P=(x_0,y_0,z_0)$ is
Cylinders
Surfaces Of Revolution
| Curve | Axis Revolved Around | Replace | By | Surface of Revolution |
|---|---|---|---|---|
| $f(x,y)=0$ | $x$ | $y$ | $\pm\sqrt{y^2+z^2}$ | $f(x,\pm\sqrt{y^2+z^2})=0$ |
| $f(x,y)=0$ | $y$ | $x$ | $\pm\sqrt{x^2+z^2}$ | $f(\pm\sqrt{x^2+z^2},y)=0$ |
And other cases are similar by replace $x$ or $y$ by $z$.
Levle Curves And Level Surfaces
For function $z=f(x,y)$, the level curve of height $c$ is given by $f(x,y)=c$.
Coordinates
Cylindrical Coordinates
The cylindrical coordinates are given by $(r,\theta,z)$ where $r=\sqrt{x^2+y^2}$ and $\theta=\arctan\frac{y}{x}$.
Spherical Coordinates
The spherical coordinates are given by $(\rho,\phi,\theta)$ where $\rho=\sqrt{x^2+y^2+z^2}$, $\theta=\arctan\frac{y}{x}$ and $\phi=\arctan\frac{\sqrt{x^2+y^2}}{z}$.
Derivatives
Partial Derivatives
Gradients
Consider the surface $z = f(x, y)$, and let $P = (x_0, y_0)$ be a point in the domain of $f$. The gradient of $f$ is the vector $\nabla f=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}\right)$.
Directional Derivatives
The rate of change of $f$ at the point $P$ in the direction of a unit vector $\mathbf{u}$, is given by $D_{\mathbf{u}}f|_P=\nabla f|_P\cdot \mathbf{u}.$
The vector $\nabla f$ points in the direction in which $f$ increases most rapidly, and the magnitude of $\nabla f$ gives the maximum rate of increase.
For a function $f (x, y, z)$, the vector $\nabla f|_P$ is perpendicular (normal) to the level curve off that contains $P$.
Similar definitions and results can be given for $\mathbb{R}^3$.
Max/Min Problems
$P_0$ is a critical point of $f(x,y)$ if $\frac{\partial f}{\partial x}\big|_{P_0}=\frac{\partial f}{\partial y}\big|_{P_0}=0.$
Let $\Delta= \text{det}(H)$.
If $\Delta>0$ and $f_{xx}(P_0)<0$, then $f$ attains a local maximum at $P_0$.
If $\Delta>0$ and $f_{xx}(P_0)>0$, then $f$ attains a local minimum at $P_0$.
If $\Delta<0$, then $f$ has a saddle point at $P_0$.
If $\Delta=0$, then no conclusion can be drawn.
Integrals
Line Integrals
Line Integrals With Respect To Arc Length
Curve
Consider a function $f(x,y)$ and a curve $C$ in the $xy$-plane. A curve given parametrically by the vector equation $\mathbf{r}=r(t)=(x(t),y(t))$ is said to be smooth, if $r’(t)$ is continuous and nonzero. And it is piecewise smooth if it is composed of a finite number of smooth curves joined at consecutive endpoints.
Line Integrals In $\mathbb{R}^2$
If $C$ is an oriented, piecewise smooth curve, we partition $C$ into $n$ segments with arc length $\Delta s_i$ and choose $P_i=(x_i,y_i)$ in each segment. Then the line integral of $f$ along $C$ with respect to arc length is defined as
where $\mathrm{d}s=\sqrt{(\mathrm{d}x)^2+(\mathrm{d}y)^2}$.
In parametric form,
where the sign depends on the parameter $t$. If $t$ increases in the positive direction on $C$, then we use the $+$ sign and otherwise we use the $-$ sign.
If $C$ is a closed curve, we usually use the notation $\oint_C$ to replace $\int_C$.
Line Integrals In $\mathbb{R}^3$
Analogously, we have
The Line Integrals Of A Vector Field
Vector Field
Let $D$ be a region of the plane on which a pair of continuous functions, $M(x, y)$ and $N(x, y)$, are both defined. Then the function $\mathbf{F}=F(x,y)=(M(x,y),N(x,y))$ in $D$ is a continuous vector field on $D$.
The Line Integrals Of A Vector Field In $\mathbb{R}^2$
Let $r(t)=(x(t),y(t))$ be a parameterization of the curve $C$. Then the line integral of the vector field $F$ along $C$ is defined as:
Fundamental Theorem
The function $F(x,y)$ is a gradient field if there is a scalar field $f$ such that $F=\nabla f$. $f$ is called a potential of $F$.
If $C$ is any piecewise smooth curve oriented from the initial point $A$ to the final point $B$, and $f$ is a continuously differentiable function defined on $C$, then:
If $F$ is a gradient field, then $F$ is conservative, that is the integral of the vector field depends only on the initial and final points of $C$. Therefore, $\oint_C \mathbf{F}\cdot\mathrm{d}\mathbf{r}=0$ for any closed curve $C$, if and only if $F$ is a gradient field.
Double Integrals
Given a function of two variables $z=f(x,y)$ and a region $S$ in the $x$-$y$ plane, the double integral of $f(x,y)$ over $S$ is defined as
which is the volume of the solid region bounded by the region $S$ and the surface $z=f(x,y)$.
Green’s Theorem
A simple closed curve is one that doesn’t cross itself between its endpoints. The positive direction of the closed curve is defined as the direction you would have to walk in order to keep the region enclosed by the curve on your _left_.
Consider a simple closed curve $C$ enclosing a region $D$, so that $C$ is the boundary of $D$. If $M(x,y
)$ and $N(x,y)$ are functions that has coontinuous partial derivatives both on $C$ and throughout $D$, then Green’s theorem says that:
It connects line integrals with double integrals in the region bounded by the line.
