math 23a: vector calculus
find equation of line given a point it passes through & knowing it has to be parallel
1) identify (x, y, z) & (a, b, c) 2) use point direction: x = x + at y = y + bt z = z + ct 3) write in coordinate form (x, y, z)
two products of vectors
1) inner/dot product 2) cross product
properties of determinants
1) interchanging 2 rows/columns results in a change of the sign of a determinant's value 2) factor scalars out of any row/column if the entire row/column contains the same value 3) if the row/column is changed by adding another row/column to it, the determinant's value is the same 4) if a = αb + βc where αβ are real numbers, a = (a1, a2, a3) = α (b1, b2, b3) + β(c1, c2, c3), then the 3x3 matrix = 0
intersection of 2 lines
1) set equal 2) solve for a t & use system of equations 3) if value of t is equal, the lines intersect
speed, velocity, acceleration
1) speed/length of velocity vector: || c'(t) || 2) velocity: c'(t) 3) acceleration: c''(t)
rules of differentiation of paths & curves
1) sum 2) scalar multiplication 3) dot product 4) cross product 5) chain
determining if a function of several variables is differentiable everywhere
1) take the derivative in terms of its variables 2) ex. if the denominator stays the same and the numerator is continuous, it is differentiable
showing a function of several variables is not differentiable
1) take the derivative in terms of its variables at the point given if any 2) use the differentiable at point x0 definition: || f(x, y) - f(0, 0) - Df(x, y) [matrix values] || / || (x, y) - (x0, y0) → 0
normalizing a vector
1) use (a / || a ||) 2) solve for || a ||
find equation of line given two coordinates (x, y, z)
1) use point point form
find 2 equations of lines given two coordinates (x, y, z)
1) v = (x2 - x1)i + (y2 - y1)j + (z2 - z1)k 2) use point point form
2x2 matrix determinant
= (a11a22) - (a12a21)
4x4 matrix determinant
= a11 | a22 a23 a24 | | a32 a33 a34 | | a42 a43 a44 | - a12 | a21 a23 a24 | | a31 a33 a34 | | a41 a43 a44 | + a13 | a21 a22 a24 | | a31 a32 a34 | | a41 a42 a44 | - a14 | a21 a22 a23 | | a31 a32 a33 | | a41 a42 a43 |
3x3 matrix determinant
= a11 | a22 a23 | | a32 a33 | - a12 | a21 a23 | | a31 a33 | + a13 | a21 a22 | | a31 a32 |
equation of plane in space
A(x-x0) + B(y-y0) + C(z-z0) = 0 where normal vector n = Ai + Bj + Ck iff Az + By + Cz + D = 0 where D = -Ax0 - By0 - Cz0
2x2 matrix
[ a11 a12 ] [ a21 a22 ]
4x4 matrix
[ a11 a12 a13 a14 ] [ a21 a22 a23 a24] [ a31 a32 a33 a34] [ a41 a42 a43 a44]
3x3 matrix
[ a11 a12 a13] [ a21 a22 a23] [ a31 a32 a33]
vector subtraction
a + (b-a) = b b + (a-b) = a - wherever arrow points to is written first in the form of (i + j + k)
equation of a vector
a = ai + aj + ak
cross product
a × b = i | a2 a3 | | b2 b3 | - j | a1 a3 | | b1 b3 | + k | a1 a2 | | b1 b2 | or a × b = | i j k | | a1 a2 a3 | | b1 b2 b3 |
inner/dot product
a • b = a1b1 + a2b2 + a3b3 - a • b = <a, b>
angle between vectors
a • b = || a || • || b || • cosθ - 1) if a • b = 0, cosθ = 90 2) if a • b ≠ 0, cosθ = (v • w) / (|| v || • || w ||)
hypocycloid
c(t) = (cos^3t, sin^3t) - traces inside a circle looks like sparkles edges ✧
basic cycloid
c(t) = (t-sint, 1-cost) - traces in the lining of a circle creating number line jump curves that look like half petals
more properties of determinants
determinant = 0 if: - 1) all elements in row/column = 0 2) there are two equal rows/columns 3) there are two proportional rows/columns 4) if a row/column is the sum/difference of another
partial derivatives
df/dx: take derivative of x values, others are constants df/dy: take derivative of y values, others are constants df/dz: take derivatives of z values, others are constants
vector
directed line segments in the plane/space with a beginning (tail) and end (head)
find traces of plane
ex. (-2x + 3y + z = 6) - 1) trace in XY plane, set z = 0 2) trace in XZ plane, set y = 0 3) trace in YZ plane, set x = 0
find equation of plane that passes through a point and is perpendicular to equation of vector
ex. (0, 0, 1), (2i + j - 3k) - 1) use form: A(x-x0) + B(y-y0) + C(z-z0) = 0 2) simplify and set to 0
find area (2x2 & 3x3 determinant)
ex. (1, 1), (0, 2) (3, 2) - 1) write three equations in the form (i + j + k) 2) solve determinant depending on 2x2, 3x3 factors
find equation of plane parallel to point that passes through origin and point
ex. (7x - 4y + 2z = -10) with point (2, -1, 3) - 1) origin: set D to 0 2) point: plug in values to determine single value answer
find equation of plane determined by 3 points
ex. P= (1, 0, -1), Q = (2, 2, 1), R = (4, 1, 2) - 1) solve for n = PQ × PR in forms (i + j + k) 2) rewrite as 3x3 determinant with i, j, k and solve for equation 3) rewrite in setup and solve: A(x-x0) + B(y-y0) + C(z-z0) = 0 using any point
distance between vectors
ex. between endpoint of vector a and b - || b - a || = √(b1-a1)^2 + (b2 - a2)^2 + (b3 - a3)^2
matrix multiplication
ex. calculate ab - 1) to find value of position in ab matrix, determine dot product of position in a with entire column in b 2) to find next neighbor value in ab matrix, move to next row value in a and multiply by next column in b 3) continue until ab matrix is filled with values
composite functions
ex. f ∘ g = f(g(x))
find equation of tangent line given point, t, and tangent vector
ex. given: (x, y, z), t = #, tangent vector equation - 1) find components of tangent line equation and plug in 2) simplify where answer contains t
given two matrices of different sizes, determine size of new matrix
ex. given: 4x1 and 1x4 matrix - 1) to get ab matrix size, multiplying (mn by np) matrix gives (mp) matrix 2) to get ba matrix size, it gives (nn) matrix 3) in this example, ab size = 4x4 and ba size = 1x1
find equation of tangent line given point, t, and being asked where a particle is at t = #
ex. given: c(t) = (x, y, z), t = # - 1) find l(# the question asks for) in the set up of a tangent line equation 2) find components and simply answer to three values
finding Df(x0) or the derivative of a vector valued function of several variables
ex. given: f(x, y) = (x^2y, ye^x, y^2x) where x0 = (1, 0) - 1) find derivatives of each function in terms of all its variables (ex. derivatives of x^2y are df1/dx1 and df1/dy) 2) create matrix with values depending on the function numbers and its variables: [ df1/dx df1/dy] [ df2/dx df2/dy] [ df3/dx df3/dy]
finding the point on graph F where tangent plane is parallel to point P
ex. given: g(x, y) = 9 - 2x^2 - y^2 at (3, 3, -18) f(x, y) = 3 - x^2 - y^2 - 1) find partial derivatives of first function and plug in given point 2) find partial derivatives of second function and set them equal to the values found after plugging in given point to find x and y 3) plug (x, y) into second function to find z for (x, y ,z) where tangent plane is parallel to
completing the square
ex. given: quadratic equation (ax^2 + bx + c = 0) convert to this form (a (x + d)^2 + e = 0) - 1) find d = (b / 2a) and e = c − (b^2 / 4a) 2) keep constant on right side of the equation
find unit vector orthogonal to two vectors
ex. i + j, j +k - 1) solve (i + j) × (j + k) using 3x3 determinant set up 2) normalize answer with (a / || a ||) 3) multiply by first equation answer
find orthogonal projection of equation onto another equation
ex. i +j, i - 2j following p = ((a • v) / (|| a ||^2)) a - 1) find (a • v) and (a • a) 2) multiply by equation a
level set
ex. x^2 + y^2 + z^2 = 1 - subset of R^3 in which f is constant
linear approximation
f(x0, y0) + [df/dx (x0, y0)](x-x0) + [df/dy (x0, y0)](y-y0)
vector valued function
function whose domain is a subset U of R^m with range in R^n if n > 1
real valued function
function whose domain is a subset of U of R^m with range in R^n if n = 1
converting: polar to rectangular coordinates
given: (r, θ) - x = rcosθ y = rsinθ
converting: cylindrical to rectangular coordinates
given: (r, θ, z) - x = rcosθ y = rsinθ z = z
converting: rectangular to polar coordinates
given: (x, y) - r = r = √(x^2 + y^2) tanθ = (y / x) = arctan(y / x)
converting: rectangular to cylindrical coordinates
given: (x, y, z) - r = √(x^2 + y^2) tanθ = (y / x) = arctan( y / x) z = z
converting: rectangular to spherical coordinates
given: (x, y, z) - ρ = √(x^2 + y^2 + z^2) θ = arctan(y / x) ϕ = arccos(z / ρ)
converting: spherical to rectangular coordinates
given: (ρ, θ, ϕ) - x = ρsinϕcosθ y = ρsinϕsinθ z = ρcosϕ where ρ ≥ 0, 0 ≤ θ < 2π 0 ≤ ϕ ≤ π
unit vectors
has norm 1 = (a / || a ||) - includes i, j, k
standard basis vectors
i: (1, 0, 0) j: (0, 1, 0) k: (0, 0, 1)
level curve
if (x, y) is a function of two variables, then the set f(x, y) = c are level curves
checking if a matrix has an inverse
if determinant = 0, no inverse if determinant does not = 0, inverse
parallel planes
if their unit vectors n1/n2 satisfy n1 = +/- n2 or if two nonzero normal vectors n1/n2 are scalar multiples of each other
section
intersection of graph and vertical plane
curvature of regular path
k(t) = (|| c''(t) × c'(t) ||) / (|| c"(t) ||^3)
norm
known as length - || a || = √(a1)^2 + (a2)^2 + (a3)^2
arc length
l(c) = integral from t0 to t1 (√(x'(t)^2 + y'(t)^2 + z'(t)^2) dt
equation of a line
l(t) = a + tv - a: passes through tip/end of vector t: all real values v: direction of vector
tangent line equation
l(t) = c(t0) + (t - t0)(c'(t0))
unit normal vector
n = (Ai + Bj Ck) / (√(A^2 + B^2 + C^2) )
orthogonal projection of v on a
p = ((a • v) / (|| a ||^2)) a
theorem: calculus of paths/limits
path c(t) = (x(t), y(t), z(t)) approaches a limit as t → # iff each component of c(t) approaches the same #
epsilon delta method
the limit of f(x) as x → x0 = b iff for every number ε > 0 there exists a δ > 0 such that for any x is contained in A satisfying 0 < || x - x0 || < δ we have || f(x) - b || < ε
epicycloid
traces outside of a circle creating flower petals
oblate spheroid
type of ellipsoid where a = b = c
point point form
x = x + (x2 - x1)t y = y + (y2 - y1)t z = x + (z2 - z1)t
point direction form of a line
x = x + at y = y + bt z = z + ct - a = (x, y, z) v = (a, b, c)
elliptic paraboloid
z = (x / a)^2 + (y /b)^2 - horizontal traces: ellipses vertical traces: upwards parabolas
hyperbolic paraboloid
z = (x / a)^2 - (y / b)^2 - horizontal traces: hyperbolas vertical traces: upwards/downwards parabolas
tangent plane
z = f(x0, y0) + [df/dx (x0, y0)](x-x0) + [df/dy (x0, y0)](y-y0)
cauchy-schwarz inequality
| a • b | ≤ || a || • || b || with equality iff a is scalar multiple of b or one of them is 0
distance from point to plane
| v • n | = ( | Ax + By + Cz + D | ) / (√(A^2 + B^2 + C^2) )
triangle inequality
|| a + b || ≤ || a || + || b ||
finding volume of parallel piped
|| a × b ||
finding area of a triangle formed by two vectors
|| a × b || / 2 - triangle is half of its parallelogram
geometry of cross products
1) area of parallelogram: || a × b || = || a || • || b || • sinθ (is the absolute value of | a1 b1 | | a2 b2 |) 2) a × b is perpendicular to a and b 3) the triple (a, b, a × b) obeys the right hand rule
finding slope of tangent line to curve
1) find derivative in terms of x and plug in the given point if any
finding the inverse of a matrix
1) find determinant of matrix and rewrite as (1/d) 2) leave the diagonal line of values as is and flip other values treating the diagonal as a mirror 3) solve matrix normally 4) multiply by matrix with correct signs beginning with: [ + - + ] and so on 5) multiply by (1/d)
verify triangle inequality given two equations in (i + j + k)
1) find | a + b |, || a ||, and || b || 2) write in the form of | a + b | ≤ || a || + || b ||
verifying cauchy-schwarz inequality given two equations in (i + j + k)
1) find | a • b |, || a ||, and || b || 2) write in the form of | a • b | ≤ || a || • || b ||
properties of cross products
1) a × b = -b × a 2) distributive law 3) i × j = k j × k = i k × i = j 4) triple product: given three vectors a, b, c, the real number is (a × b) • c where a × b is orthogonal a and b
algebraic rules of cross products
1) a × b = 0 iff a and b are parallel or a/b is 0 2) a × b = -b × a 3) a × (b + c) = ab + ac 4) (a + b) × c = ac + bc 5) (αa) × b = a × (αb) = α(a × b)
ellipsoid
(x / a)^2 + (y / b)^2 + (z / c)^2 = 1 - has radius = a its traces are ellipses a = b = c is equivalent to x^2 + y^2 + z^2 = a^2
hyperboloid
(x / a)^2 + (y / b)^2 = (z / c)^2 + d - d > 0: one sheet d = 0: cone d < 0: two sheets
