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Calculus 3
Getting started
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Three-dimensional coordinate systems
plotting points in three dimensions (10:47)
distance between points in three dimensions (10:16)
center, radius and equation of the sphere (9:57)
describing a region in three dimensional space (5:06)
using inequalities to describe the region (5:48)
Sketching graphs and level curves
sketching graphs of multivariable functions
sketching level curves of multivariable functions
matching the function with the graph and level curves
Lines and planes
vector and parametric equations of a line (6:42)
parametric and symmetric equations of a line (8:38)
symmetric equations of a line (2:58)
parallel, intersecting, skew and perpendicular lines (10:32)
equation of a plane (8:00)
intersection of a line and a plane (4:13)
parallel, perpendicular, and angle between planes (9:22)
parametric equations for the line of intersection of two planes (12:29)
symmetric equations for the line of intersection of two planes (10:43)
distance between a point and a line (8:46)
distance between a point and a plane (7:11)
distance between parallel planes (8:26)
Cylinders and quadric surfaces
reducing equations to standard form (15:02)
sketching the surface (7:59)
Limits and continuity
domain of a multivariable function (10:44)
limit of a multivariable function (6:43)
precise definition of the limit for multivariable functions (34:19)
discontinuities of multivariable functions (4:07)
compositions of multivariable functions
Partial derivatives
partial derivatives in two variables (7:25)
partial derivatives in three or more variables (5:56)
higher order partial derivatives (5:57)
Differentials
differential of a multivariable function (4:24)
Chain rule
chain rule for multivariable functions (18:04)
chain rule for multivariable functions and tree diagrams (9:31)
Implicit differentiation
implicit differentiation for multivariable functions (8:14)
Directional derivatives
directional derivatives (5:54)
Linear approximation and linearization
linear approximation in two variables (6:35)
linearization of a multivariable function (6:45)
Gradient vectors
gradient vectors (3:50)
gradient vectors and the tangent plane (4:27)
maximum rate of change and its direction (5:57)
Tangent planes and normal lines
normal line to the surface (11:15)
equation of the tangent plane (5:22)
Optimization
critical points (5:24)
second derivative test (8:52)
local extrema and saddle points (11:18)
global extrema (6:54)
extreme value theorem (33:35)
Applied optimization
applied optimization (41:51)
Lagrange multipliers
two dimensions, one constraint (25:00)
three dimensions, one constraint (8:32)
three dimensions, two constraints (14:35)
Approximating double integrals
midpoint rule for double integrals (9:15)
riemann sums for double integrals (8:32)
Double integrals
average value (6:41)
iterated integrals (32:16)
double integrals (7:15)
type I and II regions (12:01)
finding surface area (10:52)
finding volume (8:30)
changing the order of integration
Double integrals in polar coordinates
changing iterated integrals to polar coordinates (10:33)
changing double integrals to polar coordinates (12:33)
sketching area (5:35)
finding area (12:01)
finding volume (12:16)
Applications of double integrals
finding mass and center of mass (11:53)
Approximating triple integrals
midpoint rule for triple integrals (11:49)
Triple integrals
iterated integrals (10:28)
triple integrals (13:34)
average value (6:30)
finding volume (13:57)
expressing the integral six ways (17:28)
Triple integrals in cylindrical coordinates
cylindrical coordinates (3:55)
changing triple integrals to cylindrical coordinates (13:47)
finding volume (12:14)
Triple integrals in spherical coordinates
spherical coordinates (5:23)
changing triple integrals spherical coordinates
finding volume (5:47)
Change of variables
jacobian for two variables (6:00)
jacobian for three variables (10:31)
evaluating double integrals
equations of the transformation
image of the set under the transformation
Applications of triple integrals
triple integrals to find mass and center of mass (10:38)
moments of inertia (8:03)
Introduction to vectors
vector from two points (5:03)
combinations of vectors (7:43)
sum of two vectors (5:31)
copying vectors and using them to draw combinations (9:36)
unit vector in the direction of the given vector (5:28)
angle between a vector and the x-axis (8:04)
magnitude and angle of the resultant force (12:53)
Dot products
dot product of two vectors (3:22)
angle between two vectors (5:21)
orthogonal, parallel or neither (6:51)
acute angle between the lines (8:52)
acute angles between the curves (16:58)
direction cosines and direction angles (8:31)
scalar equation of a line (2:56)
scalar equation of a plane (3:49)
scalar and vector projections (7:33)
Cross products
cross product of two vectors (5:38)
vector orthogonal to the plane (9:03)
volume of the parallelepiped from vectors (6:56)
volume of the parallelpiped from adjacent edges (8:12)
scalar triple product to prove vectors are coplanar (8:47)
Vector functions and space curves
domain of a vector function (5:10)
limit of a vector function (5:52)
sketching the vector equation (10:56)
projections of the curve (16:28)
vector and parametric equations of a line segment (5:00)
vector function for the curve of intersection of two surfaces (5:37)
Derivatives and integrals of vector functions
derivative of a vector function (7:53)
unit tangent vector (6:18)
parametric equations of the tangent line (8:17)
integral of a vector function (9:00)
Arc length and curvature
arc length of a vector function (9:39)
reparametrizing the curve (7:56)
unit tangent and unit normal vectors (8:49)
curvature (11:45)
maximum curvature (13:07)
normal and osculating planes (23:22)
Velocity and acceleration
velocity and acceleration vectors (5:37)
velocity, acceleration and speed, given position (4:54)
velocity and position given acceleration and initial conditions (7:54)
tangential and normal components of the acceleration vector (13:54)
Line integrals
line integral of a curve (16:29)
line integral of a vector function (10:38)
potential function of a conservative vector field (12:57)
potential function of the conservative vector field to evaluate a line integral (13:32)
independence of path (15:48)
work done by a force field (13:00)
open, connected, and simply-connected (7:41)
Green's theorem
green's theorem for one region (8:15)
green's theorem for two regions (13:24)
Curl and divergence
curl and divergence of a vector field (12:20)
potential function of the conservative vector field, three dimensions (17:33)
Parametric surfaces and areas
points on the surface (7:20)
surface of the vector equation (10:20)
parametric representation of the surface (8:28)
tangent plane to the parametric surface (10:08)
area of a surface (10:50)
Surface integrals
surface integrals (22:32)
Stokes' and divergence theorem
stokes theorem (19:19)
divergence theorem (33:29)
Final exam
calculus 3 final exam
parametric equations of the tangent line
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