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Calculus 3

Claire Bookworm (lvl 92)

Section 1

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dot(u, v)

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Date created

Mar 14, 2020

Cards (79)

Section 1

(50 cards)

dot(u, v)

Front

|u||v|cos(theta)

Back

position vector

Front

v = p - o where p is a point and o is the origin

Back

volume of a parallelopiped

Front

|dot(a, cross(b, c))|

Back

dot(u, v') + dot(v, u') chain rule

Front

derivative of dot(u, v) & why

Back

unit vector

Front

u/|u|

Back

plane

Front

Ax + By + Cz + D = 0

Back

cross(a, b)

Front

-cross(b, a)

Back

y to cylindrical

Front

coordinate conversion of ___ = r*sin(theta)

Back

displacement vector

Front

v = q - p where p and q are points

Back

line

Front

intersection of two planes

Back

hyperboloid of one sheet (along z axis)

Front

x^2 + y^2 - z^2 = k, k > 0

Back

parametrization of a plane

Front

r(s, t) = p + sv + tw

Back

parametrization of a line

Front

r(t) = p + t*v

Back

plane (at height c)

Front

z = c in cylindrical coordinates

Back

x to cylindrical

Front

coordinate conversion of ___ = r*cos(theta)

Back

z to cylindrical

Front

coordinate conversion of ____ is z = z

Back

normal vector

Front

cross(v, w), where r(s, t) = p + sv + tw

Back

rho

Front

___ = sqrt(x^2 + y^2 + z^2)

Back

elliptic paraboloid (along z axis)

Front

x^2 + y^2 = z

Back

half-plane (bounded by z-axis)

Front

theta = c

Back

magnitude

Front

what |v| = sqrt( v(1)^2 + v(2)^2 + v(3)^2 ) represents

Back

vector

Front

result of a cross product

Back

spherical coord(inates)

Front

(rho, theta, phi)

Back

r

Front

___ = sqrt(x^2 + y^2)

Back

cylinder

Front

x^2 + y^2 = k^2

Back

component of v onto u

Front

||v|cos(theta)| or dot(v, u)/|u|

Back

level surface

Front

dot((r-p), n) = 0

Back

theta

Front

arctan(y/x) or arccos(x/r)

Back

projection of v onto u

Front

(dot(v, u)/(|u|^2))*u

Back

r to spherical

Front

coordinate conversion of _____ = rho*sin(phi)

Back

cylinder (of radius c)

Front

r = c in cylindrical coordinates

Back

scalar

Front

result of a dot product

Back

y to spherical

Front

coordinate conversion of ___ = rho*sin(phi)sin(theta)

Back

cone (along z axis)

Front

x^2 + y^2 = z^2

Back

|cross(u, v)|

Front

|u||v|sin(theta)

Back

phi

Front

arccos(z/rho)

Back

parabolic cylinder

Front

y = x^2

Back

sphere (of radius c)

Front

rho = c

Back

theta

Front

angle in xy-plane starting from positive x axis

Back

z to spherical

Front

coordinate conversion of ___ = rho*cos(phi)

Back

|v|^2

Front

dot(v, v)

Back

x to spherical

Front

coordinate conversion of ___ = rho*sin(phi)cos(theta)

Back

cone (bounded by xy-plane)

Front

phi = c

Back

phi

Front

drop angle from positive z axis

Back

ellipsoid

Front

x^2 + y^2 + z^2 = k, k > 0

Back

cylindrical coord(inates)

Front

(r, theta, z)

Back

rho

Front

distance to the origin in spherical coordinates

Back

area of a parallelogram

Front

|cross(u, v)| where u is adjacent to v

Back

hyperbolic paraboloid (along z axis)

Front

x^2 - y^2 = z

Back

hyperboloid of two sheets (along z axis)

Front

-x^2 - y^2 + z^2 = k, k > 0

Back

Section 2

(29 cards)

v = -n

Front

how to find v for a line parallel to a plane with normal vector n

Back

arc length

Front

integral[|r'(t)|dt, t = a, t = b]

Back

0 <= phi <= pi

Front

range of phi values for spherical coordinates

Back

rho^2*sin(phi)

Front

volume stretching factor of rectangular to spherical coordinates

Back

z' = fx(x0)(x - x0) + fy(x0)(y - x0) + f(x0)

Front

tangent plane formula for z = f(x,y) and x0 = (x0, y0)

Back

elliptic paraboloid (along y axis)

Front

x^2 + z^2 = y

Back

check if dot(L1, L2) != 0

Front

how to find if there are no planes containing L1 perpendicular to L2

Back

hyperboloid of one sheet (along x axis)

Front

-x^2 = k - z^2 - y^2, k > 0

Back

linear approximation

Front

L(x0, y0) = f(x0, y0) + (df/dx)(x0, y0)(x - x0) + (df/dy)(x0, y0)(y - y0)

Back

flowing inward

Front

div(F) is negative

Back

0 <= theta <= 2pi

Front

range of theta values for spherical and cylindrical coordinates

Back

hyperboloid of two sheets (along x axis)

Front

x^2 - z^2 = k + y^2, k > 0

Back

det(dx/du, dx/dv; dy/du, dy/dv)

Front

jacobian for converting (x,y) to (u, v)

Back

(v = )cross(n1, n2)

Front

how to find v for parametrization of a line given by intersection of two planes with normal vectors n1 and n2

Back

continu(ity)

Front

when lim(x,y) -> (a, b) of f(x, y) = f(a, b)

Back

let x(s, t) = s, y(s, t) = t, z(s, t) = (x(s, t), y(s, t), D)

Front

how to convert A(x - x0) + B(y - y0) + C(z - z0) = 0 to parameterization, where D = Ax0 + By0 + Cz0

Back

hyperbolic paraboloid (along y axis)

Front

x^2 - y = z^2

Back

hyperbolic paraboloid (along x axis)

Front

x = -z^2 + y^2

Back

hyperboloid of one sheet (along y axis)

Front

x^2 + z^2 = k + y^2, k > 0

Back

dot(r'(t), r''(t)) + dot(r''(t), r'(t))

Front

derivative of dot(r'(t), r'(t))

Back

cone (along x axis)

Front

x^2 - y^2 - z^2 = 0

Back

cone (along y axis)

Front

x^2 - y^2 + z^2 = 0

Back

flowing outward

Front

div(F) is positive

Back

limit DNE

Front

if the limit of two curves approaching the same point are not equal

Back

hyperboloid of two sheets (along y axis)

Front

y^2 = k + x^2 + z^2, k > 0

Back

elliptic paraboloid (along x axis)

Front

-x + y^2 = -z^2

Back

shadow of b cast (straight down) on a

Front

geometrically what projection of b onto a represents

Back

r(t) + s*r'(t)

Front

parametrization of the tangent line to a curve at some point t for curve of r(t)

Back

height

Front

if z = f(x, y) then z is the _____ of some point (x, y) in the xy-plane

Back