Calc 3 Chapter 12

how to represent x,y coordinates on a circle with radius R using cos and sin?

(Rcos(t), Rsin(t)) = Rcos(t)i + Rsin(t)J

circle centered at 0,0 using x and y formula

x2 + y2 = r2

center shifts w/ cos and sin

2 + cost would be if the center is at x=2

cost - 2 would be if the center is at x=-2

r(t) = *as a coordinate

(x(t), y(t), z(t))

don’t forget the constant of integration (which is a vector meaning that it adds a constant c₁ c₂ and c₃ to the ijk respectively. the constants can be different so you need to solve for them!)

what is a zero vector

all the values for (x,y,z) are (0,0,0)

speed =

magnitude of velocity aka take derivative of the position function then take norm of that

what does the vector r’(t) represent if r(t) is continuous and differentiable

r’(t) is a vector tangent to the motion of the curve traced by r(t)

A curve is closed if it…

A curve is closed if it has all of the following:

• a parameterization r^→(t)

• a ≤ t ≤ b

• r^→(a) = r^→(b)

A curve is smooth if…

A curve if smooth if for SOME parameterization r(t) along [a,b]

• r’(t) is continuous & exists for all t in the interval [a,b]

• r’(t) ≠ 0^→ for all t in the interval [a,b] aka r’(t) isn’t a zero vector for all t in [a,b]

e.g. a circle

piecewise smooth defn

Piecewise smooth means that the curve can be divided into a finite number of sub intervals like [a,b] [b,c] [c,d] etc where each sub interval is smooth

e.g. a square has 4 corners making it not differentiable in those spots, but the square can be broken up into pieces where each piece is a line that is smooth within its subinterval

how to obtain new parameterizations (e.g. r(t) = (t, t²)

you can do t = t² and get a new parameterization, q(t) which = (t², t⁴)… essentially you can plug in different values

arclength formula + what types of curves does it work for?

aka integrate speed aka integrate the magnitude of velocity

• works for smooth and piecewise smooth curves

• only works if r(t) traces the curve only once within [a,b]

Unit tangent vector (formula)

normalize velocity

What is a principal normal vector?

The principal normal vector is a vector of length one that is perpendicular to a curve and points toward the inside of the curvature

Note that the principal normal vector isn’t defined at inflection points bc there is no curvature so pointing “inwards” doesn’t exist

Straight lines have no curvature and therefore the principal normal vector is undefined

principal normal vector formula

normalize the derivative of the unit tangent vector

what does the tangential component of acceleration control? what does the normal component of acceleration control?

a_TT^→ changes a particle’s speed

a_NN^→ changes a particle’s direction

a_N is ____ to 0 because ____

the normal component of acceleration is always greater than or equal to 0 because the direction can’t be negative

tangential component of acceleration formula

[velocity dotted with acceleration] all over the magnitude of velocity

normal component of acceleration formula

the magnitude of the [velocity crossed with acceleration] all over the magnitude of velocity

a^→(t) =

where:

• a_T and a_N get evaluated as the tangential and normal components, respectively, of acceleration

• the symbols T^→ and N^→ stay as they are written as if were variables such as x and y

How to use Pythagorean theorem to get the tangential or normal component of acceleration easier (FORMULA)

how to find where a plane and some parameterization r(t) written in ijk form intersect

plug in the different components from r(t) into x,y, and z in the plane equation

e.g.

r(t) = 5t²i - (t+1)j + (t²+1)k

8x+7y+0z = 5

8(5t²) + 7( -(t+1) ) + 0(t² + 1) = 5 now solve for t