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#rs/notes #rs/class/math163
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- - -
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- 3D lines
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- can be described as parametric
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- symmetric equation of 3D line $$ \frac{x-x_{0}}{a} = \frac{y-y_{0}}{b} = \frac{z-z_{0}}{c}$$
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- In this case, the direction vector is $<a, b, c>$
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- Distance between a point and a line
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- Is the shortest distance between the two
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- This is the segment perpendicular to the line
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- In 2D, you have to take random point on the line
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- then project vector from this point to point not on the line onto the line to find where the closest point on the line is
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- ![[Pasted image 20251014183240.png]]
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- In this parallelogram, we know the area is equal to either $v \times PM$ or the base * the height (the height is the distance beween M and the line
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- This gives the formula $$d=\frac{||PM \times v ||}{||v||}$$ For the distance between point M and the line
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- Lines in 3D
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- Can either be parallel, intersection, the same line or skew
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- skew is when they don't intersect but aren't parallel either
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- Planes in 3D
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- defined by the normal vector of the plane
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#rs/notes #rs/class/math163
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- - -
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-
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#rs/notes #rs/class/math163
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- - -
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-
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#rs/notes #rs/class/math163
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- - -
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- Similar idea for multiplication, cross products and dot products with derivatives $$ \frac{d}{dt} [r(t) \times u(t)] = r'(t) \times u(t) + r(t) \times u'(t)$$ $$ \frac{d}{dt} [r(t) \cdot u(t)] = r'(t) \cdot u(t) + r(t) \cdot u'(t)$$
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- If $r(t) \cdot r(t) = c$, then $r(t) \cdot r'(t) = 0$
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- Principle unit tangent vector is defined as $$T(t) = \frac{r'(t)}{||r'(t)||}$$ as long as $||r'(t)|| \ne 0$
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- When there is a function of two variables with all real numbers as domain, it is written as $\mathbb{R}^2$
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- Can also write "domain: $\{(x,y) | x^2+y^2 \leq 9\}$"
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#rs/notes #rs/class/math163
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- - -
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- Level curve
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- A 2D shape that results from setting a two-variable equation equal to a constant
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- can help to see the shape of a 3D function
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- basically like taking horizontal slices out of a two-variable function
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- Vertical trace
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- similar to level curve, finding the 2D shape from a vertical slice
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- can find with either $f(a, y) = z$ for constant $x = a$ or $f(x, b) = z$ for constant $y=b$
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- Both of these are used to visualize 3D functions in 2D
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- Functions of 3 variables
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- there is no great way to visualize these in 3 dimensions
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- have to take 3D "slices" out of the 4D shape
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#rs/notes #rs/class/math163
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- - -
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- Direction angles
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- angles vector forms with axis
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- vector projection
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- Projection of one vector onto another: $$proj_{u}v=\frac{u\cdot v}{\left|\left|u\right|\right|^{2}}u$$
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- Vector that is same direction
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- To just get the magnitude of the projected vector, use: $$ mag_{u}v=\frac{|u*v|}{||u||^2} $$
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- Unit vectors
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- magnitude is one
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- Formula: $$ unit_u=\frac{u}{||u||}$$
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- Resolving vectors to components
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- project one vector to another
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- subtract projection from original vector
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- Determinate
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- equation: $$\begin{vmatrix} a&b\\c&d\end{vmatrix}\rightarrow ad-bc$$
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- larger than 2x2: $$\begin{vmatrix} a&b&c\\d&e&f\\g&h&i\end{vmatrix}\rightarrow a\begin{vmatrix} e&f\\h&i\end{vmatrix} - b\begin{vmatrix} d&f\\g&i\end{vmatrix} + c\begin{vmatrix} d&e\\g&h\end{vmatrix}\rightarrow a(ei-hf)-b(di-gf)+c(dh-ge)$$
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- Cross product
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- creates vector that is orthogonal to both vectors
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- for which direction it goes, use right hand rule
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- pointer finger is first vector, middle is second
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- thumb is resulting vector
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- equation: $$a \times b = \begin{vmatrix} i&j&k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}= i(a_2b_3-a_3b_2)-j(a_1b_3-a_3b_1)+k(a_1b_2-a_2b_1)$$
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- cross product is not communitive $$u\times v \ne v \times u$$
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- it is anti-communitive though $$ u \times v = -(v \times u) $$
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- this too $$ a \times a = 0 $$
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#rs/notes #rs/class/math163
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- - -
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- magnitude of the cross product
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- equation: $$||u \times v|| = ||u|| * ||v|| * \sin(\theta)$$
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- area of a parallelogram
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- if there are two (2D) vectors that are two adjacent sides, the area is $$||a \times b||$$
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- triple scalar product: $$u * (v \times w) = \begin{vmatrix} u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \\ u_{1} && u_{2} && u_{3} \end{vmatrix}$$
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- (can either take the determinate of the matrix or the dot product of the cross product)
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- results in a scalar
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- This is equal to the volume of the parallelepiped with the three vectors representing adjacent edges
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- $$u * (v \times w) = (u \times v) * w $$
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- lines and planes in 3D space
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- can use 3D vectors to describe 3D lines
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- $PQ = tv$
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- $<x-x_{0}, y-y_{0},z-z_{0}> = t<a,b,c>$
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- $<x_{0}, y_{0}, z_{0>}$ is the initial point of the line (or just a point on the line)
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- $<a,b,c>$ is the 3D slope of the line
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- $t$ is the independent variable
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- can also write it as $$<x, y, z> = t<a,b,c> + <x_{0}, y_{0}, z_{0}>$$
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-
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#rs/notes #rs/class/math163
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- - -
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- Equation for a normal plane to a curve: $$z = f(x_{0}, y_{0}) + f_{x}(x_{0}, y_{0})(x-x_{0}) + f_{y}(x_{0}, y_{0})(y-y_{0})$$
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-
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#rs/notes #rs/class/math163
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- - -
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- Chain rule
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- If there is a function $z(x(t), y(t))$ then $\frac{dz}{dt} = \frac{dz}{dx} \frac{dx}{dt} + \frac{dz}{dy} \frac{dy}{dt}$
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- If $z=f(x(u,v), y(u,v))$ then $$\frac{{\partial z}}{\partial u} = \frac{{\partial z}}{\partial x} \frac{{\partial x}}{\partial u} + \frac{{\partial z}}{\partial y} \frac{{\partial y}}{\partial u}$$ $$\frac{{\partial z}}{\partial v} = \frac{{\partial z}}{\partial x} \frac{{\partial x}}{\partial v} + \frac{{\partial z}}{\partial y} \frac{{\partial y}}{\partial v}$$
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- Implicit differentiation
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- If $z$ is defined implicitly as a function of $x$ and $y$, then $$ \frac{dz}{dx} = - \frac{{\frac{{\partial f}}{\partial x}}}{\frac{{\partial f}}{\partial z}}$$ $$ \frac{dz}{dx} = - \frac{{\frac{{\partial f}}{\partial y}}}{\frac{{\partial f}}{\partial z}}$$
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- Critical points
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- For functions of two variables, this is when they both equal 0 or when one is undefined
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- Second derivative test: $$D = f_{x x}(x_{0}, y_{0})f_{y y}(x_{0}, y_{0}) - (f_{x y}(x_{0}, y_{0}))^2$$
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- if $D>0$ and $f_{x x}(x_{0}, y_{0})>0$ then $f$ has a local minimum at $(x_{0}, y_{0})$
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- if $D>0$ and $f_{x x}(x_{0}, y_{0})<0$ then $f$ has a local maximum at $(x_{0}, y_{0})$
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- if $D<0$ then $f$ has a saddle point at $(x_{0}, y_{0})$
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- if $D=0$ then the test is inconclusive
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#rs/notes #rs/class/math163
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- - -
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- Double integrals
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- used to find volume underneath 3D curve $$\int \int f(x, y)dA = \lim_{ m,n \to \infty } \sum_{i=1}^m \sum_{j=1}^n f(x_{i}^*, y_{j}^*)\Delta A$$
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- Properties
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- sum: $$\int \int [f(x, y) + g(x, y)]dA = \int \int f(x, y)dA + \int \int g(x, y)dA$$
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- constant: $$\int \int cf(x, y)dA = c\int \int f(x, y)dA$$
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- Iterated integrals
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- the iterated integral for a function $f(x, y)$ over the rectangular region $R = [a, b] \times [c,d]$ is $$\int_{a}^b \int_{c}^d f(x, y)dy \ dx = \int_{a}^b\left[ \int_{c}^d f(x, y) dy\right]dx$$
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- Fubini's theorem
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- if a function is continuous over the region, then the double integral equals the iterated integral: $$\int \int f(x, y) dA = \int \int f(x, y)dx \ dy = \int_{a}^b \int_{c}^d f(x, y) dx \ dy = \int_{c}^d \int_{a}^b f(x, y) dy \ dx$$
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#rs/notes #rs/class/math163
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- - -
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- average value of a funciton of two variables over a region R is: $$f_{ave} = \frac{1}{Area \ R}\int \int_{R} f(x, y) dA$$
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-
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#rs/notes #rs/class/math163
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- - -
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- Vectors
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- have both magnitude and direction
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- Operations on vectors
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- scalar multiplication
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- multiply both components of vector by the same scalar
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- doensn't change the direction of the vector
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- addition
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- add the corresponding components together
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- same as putting initial point of one on terminal point of other
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- subtraction
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- v - w can be represented as v + (-w)
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- when initial points are the same, difference is the vector formed between the two terminal points
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- can also add the negative of one vector to the other
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- Component form
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- when a vectors initial point is at (0,0) it can be written in component form
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- <x, y>
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- Magnitude
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- Pythagoras
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