CALC III
Principle unit normal vector N
(dT/dt)/|dT/dt| because the derivative of the unit tangent vector is always normal to T, so you just want the unit vector of that. N always points in the direction in which T is turning.
Rectangular coordrinates
(x,y,z)
U X V
(|u||v|sin(theta)) n remeber n, its the direction if the two vectros are parallel, the cross product is zero because sin 0 = 0 OR JUST USE DETERMINANT IF WE DONT HAVE ANGLE
A function is continous at a point (x0, y0) if
1. f is defined at (x0, y0) 2. limit at that point exists limit = f(x0,y0)
Properties of directional derivatives
1. the function f increases most rapidly when grad f and direction vector u are in in the dame direction (ig cos theta = 1 or theta = 0). 2. f decreases most rapidly when the theta is pi 3. If the direction vector u is orthogonal is the gradient, then f does not change in that direction because the directional derivative is 0.
A line can be defined with
A point and a vector parallel to the line
Define continuity for a vector valued function
A vector valued function is continous at a point t=t0 if lim as t approaches t0 of r(t) is equal to r(t0). A function is continous if it is continous over the interval of its domain.
If parametrizing line segement as opposed to line, make sure to
Add restrictions for t such that it gos from the first point to the last point. eg. 0<_ t<_1
Arc length Parameter S(t)
Basically getting arc length function from a given reference point to t. Returns function, not value
Vector operations properties
Basically, anything goes because you're just adding, subtracting and multiplying. Thus, they are distributive and stuff. The order does not matter.
Why do streams flow perpendicular to contour curves
Because grad is perpendicular to level curves, and the fastest rate of change is in the direction of the grad.
Unit vector T
CALLED UNIT TANGENT VECTOR. Its the direction vector of the derivative (mag 1). When finding it, unless its at a specifc point, youll have Ts in it.
Standard Equation for a sphere
Comes from distance formula because the distance to the points will always be the radius (x-x0)^2+(y-y0)^2+(z-z0)^2 = a^2 where x0 is the center of the circle
Finding angle between vectors
Do dot prduct and sole for theta
When working with tension problems
Do the sum of forces equals zero and solve for forces.
(x^2/a^2) + (y^2/b^2) + (z^2/C^2) = 1
Ellipsoid
To find angle between two planes
Find angle between their normals.
Right hand rule for axis
Fingers curl from positive x to positive y. Thumb is the direction of z.
To find the point where a line intersects a plane
For a line to intersect a plane, the point of intersection must satisfy the equation of the plane. This, plug the parametrized x, y, and z equations of the line into the plane equation and solve for t. Then, plug into the line euqation.
To find the point in which a line interesects a plane
For aoint to be on the plane, it must p
Component functions
Functions of the components of vector valued functions
To find the line that intersects to planes
Get the cross products of their normals and then use that and a point on the line to find the parametric equation of the line. To find a point of the line, solve a system of equations the the equation of the planes in which u plug in zero (or any point really) for a component of both equations.
dr/ds
How position changes with respect to arc length. It is equal to T (direction vector of dr/dt)
A vector valued functions has a derivative
If its component functions all have derivatives. The derivative is the vector made from the derivative of the component functions.
In what direction does a function decrease most rapidly
In the direction opposite of gradient, so just find the direction vector of the gradient and multiply it by (-1)
Circle of curvature (AKA osculating cirle)
Is a circle made at every point p in a curve that has the same curvature as the curvature at point p.
If a vector function has constant length then
It and its derivative are always perpendicular (think motion around sphere) and thus r dot dr/dt = 0
A domain is closed if
It contains the boundary points (AKA the last point AKA [])
A function is differentiable if
It is diferentiable at everypoint
Distance from a point to a line in space
Its just the scalar component of the vector PS in the direction of the normal of V, so |PS|sin(theta) which is |PS X v|/|v| Where v is parallel to the line segement
If asked for vector perpendicular to plane
Its the cross product of two vectors on the plane
Adding vectors
Just add components. This gives the resultant vector
Midpoint of a line segment from P1 to P2
Just take averages. (X1+X2/2, y1+Y2/2, z1+z2/2)
Chain rule
Just take df/dx * dx/dt for every variable and add them.
To find the normal of a plane
Just use the coefficients of the plane equation. Its literally the reverse of getting a plane equation from the normal.
Length of a smooth curve
L = a to b ∫sqrt(sum of derivatives of each component (each squared)) Think of it as adding the magnitudes of the tangents
Speed
Magnitude of velocity
Multiplying a vector by a scalar
Multiply every component by the scalar
Difference between limits with two variables and limits with 1
Now, because the domain is a plane, we can approach a point from any direction.
Questions about TNB
Often ask you to write acceleration in the terms above
flittle x and f little y
Partial derivatice with repect to x and y
To tell whether a region is bounded, (talking about domain)
Picture if it could fit on a finite disk
When attempting to sketch a level curve, say z = 6
Plug in 6 for z and try to picture the graph in terms of x and y.
Projection of U onto V
ProjvU. This represents the value of U in the direction of V
torque vector
R X F n or |R||F|sin theta n Points according to the right hand rule
Indefinite integral of a vector valued function r(t)
R(t) + C
epsilon delta limit definition with two variables
Same as before, except now 0 < sqrt[(x-x0)^2 + (y-y0)^2] < delta because that's the distance from the chosen x value to the x value specified by the limit.
Finding derivative of a cross product
Same as if where multiplication, except with cross product
Finding derivative of a dot product
Same as if where mutliplication, except with dot product.
How to find definite integral of a vector valued function
Take the definite integral of each of the components. RETURNS A CONSTANT VECTOR.
Partial derivative dz/dx of function where z defines a function of x and y
Take the derivative of the entire function, with y constant, x the derivative thing, and also differentiate z, but add dz/dx to it.
How to find indefinite integral of a vector valued function
Take the integrals of each component, add the ijk and then add all their Cs into one big C vector at the end. GIVES VECTOR FUNCTION
Limits for vector valued functions
Take the limit of each component. Result is a vector
A function is smooth if
The derivative is continuous and never 0, that is the derivatives of the component functions are never all simultaneously 0.
Parametric Equation for a line
The parametracion of the line through P0 parallel to V = v1 i + V2 J + V3 K is x = x0 + tv1 y = y0 + tv2 z = yo+tv3
Distance from a point to a plane
The scalar component of PS in the direction of n. Thus, |PS dot n / n| P is a point on the plane. To find it just find any point that satisfies the plane equation or just set the other values to zero (intercept)
Level surface
The set of points (x,y,z) in space where a function f(x,y,z) has a constant value C. Forms a surface
Level curve of f
The set of points where f(x,y) has a constant value C.
How to tell if two vectors are parallel
Their cross product is 0
Two vectors are orthoganol if
Their dot product is zero, meaning the angle between them is pi/2
Two planes do not intersect if and only if
Their normals are parallel
Two vectors are equal if
They have the same length and direction
Shortcut for implicit differentiation
To find say dy/dx of an implicitly defined function more easily, you can just find the partial derivative with respect to x and divide it by the partial derivative with respect to y. MUST BE AN EQUATION THAT EQUALS 0.
Direction vectors are often called
Unit vectors. Distinguish them from i,j,k because they are the unit vector of an entire vector.
Work done by a constant force
W = F dot d because its the scalar component of F in the direction of D times the length of D, which gives the above equation
Domain exclusions
We exclude numbers that lead to complex (imaginary) numbers or division by zero
limit operations
Whatever is done to the function can be done to the functions limit.
Partial derivative
When all but one independent variable get held constant.
When does flittlexy = flittleyx
When the derivatives and the function are continuous.
Implicit differentiation
When the variable you are trying to derive is not equal to a function, you must derive the entire function by the bottom variable and when u get to your target one, derive it with respect to it, but the add dy/dx or whatever.
Coplanar
When three or more vectors lie in the same plane. Such as u,v, u+v
Properties of dot product
You can do everything because its just multiplication and addition. Also, U dot U is|u^21
Properties of cross Product
You can factor and u can distribute. You cant change order Two important ones 1. V X U = -(U X V) 2. U X (V X W) = (U dot W)V - (U dot v)W
Chain rule with two independent variables
You gotta do to derivatives, one dw/ds and the other dw/dr where r and s are basically like ts.
How to define a plane
You need a point and the vector normal the the plane. Because a plane is made up of thousands of vectors that are orthagonal to the same normal, the dot product of the vectors should be zero. Which is why for a planes normal vector Ai + Bj + Ck and a vector on the plane between two points PoP, the plane equation is A(x-x0) + B(y-y0) + C(z-z0) = 0
Graphical understanding of partial derivative
You slice the graph with the plane of the variable you held constant and the find the slope of the slice. In other words, how f changes with x when y is constant or vice versa.
Only vector with no direction
Zero vector
Combining the length and the direction of a projection gives us this formula for the projection of U onto V.
[(U dot V)/|V^2|] * V
If the acceleration vector is written as a = atT + anN, then
aT = d/dt * |v| and aN = k |v^2|
For projectile motion problems remeber this
acceleration will always be -gj (use to get integrals) You will also have to get total intial velocity (add external) to find equations
Tangent line for a level curve
because the dot product of two perpendicular vectors is zero, we use this with the gradient at (x0,y0) and a line that passes through (x0,y0) to find that flittlex(x0,y0)(x-x0) +flittley(y-y0) = 0
If a function is differntiable at a point, it is also
continous
Domain and range of sin(xy)
d: entire xy plane r: [-1,1]
To show that a limit does not exist
demonstrate that there are two paths that lead to a different limit. Basically, first do y =mx, if answer depends on m ur done. if not, do x =cy^2. If answer depends on C, ur done.
Directional derivative DuF
df/ds u,po = grad(f)po dot u (u is direction vector). Think about it as putting the derivative in vector form and then putting it in the direction we want.
If the ask you to find the unit vector of a vector, its the
direction vector
When a function has the same limit along all straight lines approaching (x0, y0)
does not imply a limit exists
With scalar functions with more than 1 independent variable, the domain is the plane formed by the dependent variable
eg. z = x +y , the domain is the xy -plane
How to find direction of zero change
find the normals of the direction of the gradient (i+j), which are (-i +j) and (i-j)
Algebra for gradients
grad (f+g) = grad f + grad g (same for difference) grad (kf) = k grad f grad (fg) = fgrad(g) + ggrad(f) grad(f/g) =[ ggrad(f)-fgrad(g)]/g^2
Gradient Vector
gradient of f(x,y) at the point p(x0,y0) is df/dx i + df/dy j
When working with unit vectors leave answers in terms of
i,j,k
In what direction does a function increase most rapidly
in the direction of the gradient, so just find the direction vector of the graditent.
A domain is open if
it doesnt contain boundary points (AKA the last point AKA ())
A function is unbounded if (talking about domain)
it is not bounded
A function is bounded if (talking about domain)
it lies inside a disk of finite radius.
The scalar component of U in the direction of V
its literally just the length of the projection, which is |U|cos(theta) but can be rearranged to (U*V)/|V| Its basically the projection without the direction
When something is like this (...] or it has part of its boundary missing.
its neiter opened nor closed
How to find a vector parallel to a line segment
just get the coefficients of the ts and add them to i , j, k duhhhhh
Because we cant really visualize 4th degree functions, we observe their
level curves and how they change.
Curvature function (k) of a curve
mag(dT/ds) cuase its how its direction changes with length. More easily put, its k = 1/|v| * |dT/dt|
Vector vs line
many vectors point to points on a line
Radius of the circle of curvature
p = 1/k
The cross product gives you the area of a
parallelogram. Divide by two to find area of triangle
Orthogonal synonym
perpendicular
gradients are what to level curves
perpendicular because there is no change in that direction.
Ideal Projectie motion equation
r = (V0cosa)ti + [(V0sina)t-(1/2)gt^2]j where alpha is the launch angle
Vector valued functions formula
r(t) = f(t)i + g(t)j + h(t)k
Vector Equation for a line
r(t) = r0 + tv where r0 is the position vector of a point on the line Also written as r0 + t|v|(v/|v0) It basically says that the position of the point on a line is equal to the initial position plus the speed times time in the direction of V
U-V
same as U+(-V) = <u1-v1, u2-v2, u3-v3)
Distance between two points
sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)
Magnitude of a vector
sqrt(v1^2+v2^2+V3^2)
aN also equals
sqrt(|a|^2 -aT^2)
Derivative along a path
supposing r(t) is a vector path and w = f(r(t)) is a function evaluated along the path, then dw/dt = grad(f(r(t)) dot r´(t)
With chain rule, always leave final answer in terms of
the indepependent variable( eg t,r,s)
Conotu curves are basically
the same as level curves
The direction of motion is always the direction of
the velocity vector. Thus to get direction of motion, get the direction vector of the velocity.
U dot V
u1v1 + u2v2 + u3v3 |u||v|cos(theta) where theta is angle between the vectors.
Component form of a vector
v = <v1,v2,v3>
The direction of a projection of U onto V is
v/|v|
The veloicty of a plane in the air with respect to the ground is given by
velocity of the plane + velocity of the air. Add vectors.
Vector Valued Functions
x = f(t), y = g(t), z = h(t) Produces points (or another way to think of it is vectors from the origin) because both magnitude and direction change, the derivatives and integrals are also vectors
Triple Scalar Product
|(U X V) dot W| Gives the volume of a parallelopiped with base U X V and height W Also |U X V||W||costheta| OR JUST THE MAGNITUDE OF THE DETERMINANT OF THE ORDER U, THEN V, THEN V
Magnitude of a cross product
|U||V|sin(theta) or just take the magnitude of the formed cross product.
The length of a projection of U onto V is
|u|cos(theta)
