Calculus III - Multivariable Calculus, Vector Analysis, and Differential Equations

Krista King via YouTube

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Master advanced calculus concepts through this comprehensive video course covering multivariable calculus, multiple integrals, vector analysis, and differential equations. Explore limits and continuity of multivariable functions, then delve into partial derivatives, tangent planes, normal lines, and linear approximation techniques. Learn to apply the chain rule and implicit differentiation to multivariable functions, work with directional derivatives and gradient vectors, and solve optimization problems using Lagrange multipliers. Study double and triple integrals with applications to finding volumes and surface areas, including conversions to polar, cylindrical, and spherical coordinate systems. Develop proficiency in three-dimensional coordinate systems, vector operations including dot and cross products, and equations of lines and planes in space. Analyze vector functions, space curves, arc length, curvature, velocity, and acceleration, then progress to line integrals, Green's theorem, curl and divergence, parametric surfaces, and surface integrals. Complete your study with differential equations, covering linear, separable, and exact equations, population modeling techniques, Euler's method for numerical solutions, second-order homogeneous and nonhomogeneous differential equations, and Laplace transforms for solving initial value problems.

Syllabus

domain of a multivariable function (KristaKingMath)
limit of the multivariable function (KristaKingMath)
precise definition of the limit for multivariable functions (KristaKingMath)
discontinuities of a multivariable function (KristaKingMath)
Partial Derivatives (KristaKingMath)
Second Order Partial Derivatives (KristaKingMath)
Equation of the Tangent Plane in Two Variables (KristaKingMath)
Normal line to the surface (KristaKingMath)
Linear Approximation in Two Variables (KristaKingMath)
Linearization of a multivariable function (KristaKingMath)
Differential of the multivariable function (KristaKingMath)
Chain rule for partial derivatives of multivariable functions (KristaKingMath)
Chain rule and tree diagrams of multivariable functions (KristaKingMath)
Implicit differentiation for partial derivatives of multivariable functions (KristaKingMath)
Directional Derivatives (KristaKingMath)
Gradient Vectors (KristaKingMath)
Gradient Vectors and the Tangent Plane (KristaKingMath)
Gradient vectors and maximum rate of change (KristaKingMath)
critical points of multivariable functions (KristaKingMath)
Second Derivative Test - Two Variables (KristaKingMath)
Local extrema and saddle points of a multivariable function (KristaKingMath)
Global Extrema in Two Variables (KristaKingMath)
Extreme value theorem, extrema in the set D (KristaKingMath)
Max product of 3 real numbers (KristaKingMath)
Max volume of a rectangular box inscribed in a sphere (KristaKingMath)
minimum distance between the point and the plane (KristaKingMath)
Points on the cone closest to a point (KristaKingMath)
Lagrange Multipliers PART 1/2 (KristaKingMath)
Lagrange Multipliers PART 2/2 (KristaKingMath)
lagrange multipliers, three dimensions one constraint (KristaKingMath)
Lagrange multipliers in three dimensions with two constraints (KristaKingMath)
Midpoint rule to approximate volume of a double integral (KristaKingMath)
Riemann sums to approximate volume of a double integral (KristaKingMath)
Average value of a double integral (KristaKingMath)
Iterated Integrals (KristaKingMath)
Iterated integrals (KristaKingMath)
Double integrals (KristaKingMath)
Double integrals of type I and type II regions (KristaKingMath)
Double integrals to find the volume of the solid (KristaKingMath)
Double integrals to find surface area (KristaKingMath)
Converting iterated integrals to polar coordinates (KristaKingMath)
Converting double integrals to polar coordinates (KristaKingMath)
Sketching the region given by a double polar integral (KristaKingMath)
Double polar integral to find area (KristaKingMath)
Double polar integral to find the volume of the solid (KristaKingMath)
Double integrals to find mass and center of mass of the lamina (KristaKingMath)
Midpoint rule for triple integrals (KristaKingMath)
Triple iterated integrals (KristaKingMath)
Triple integrals (KristaKingMath)
Average value of the triple integral (KristaKingMath)
Triple integrals to find volume of the solid (KristaKingMath)
Expressing a triple iterated integral 6 ways (KristaKingMath)
Mass and center of mass with triple integrals (KristaKingMath)
Moments of inertia with triple integrals (KristaKingMath)
Cylindrical Coordinates (KristaKingMath)
Converting triple integrals to cylindrical coordinates (KristaKingMath)
Volume in cylindrical coordinates (KristaKingMath)
Spherical Coordinates (KristaKingMath)
Triple integral in spherical coordinates to find volume (KristaKingMath)
Jacobian of the transformation (2x2) (KristaKingMath)
Jacobian of the transformation (3x3) (KristaKingMath)
Plotting points in three dimensions (KristaKingMath)
distance between two points in three dimensions (KristaKingMath)
Equation of a sphere, plus center and radius (KristaKingMath)
Describing a region in 3D space (KristaKingMath)
Using inequalities to describe a region in 3D space (KristaKingMath)
Finding a Vector From Two Points (KristaKingMath)
Vector addition and combinations of vectors (KristaKingMath)
Sum of Two Vectors (KristaKingMath)
Copying vectors to find combinations of vectors (KristaKingMath)
Unit vector in the direction of the given vector (KristaKingMath)
Angle between a vector and the x-axis (KristaKingMath)
Magnitude and angle of the resultant force (KristaKingMath)
Dot product of two vectors (KristaKingMath)
Angle between two vectors (KristaKingMath)
Orthogonal, parallel or neither (vectors) (KristaKingMath)
Acute angle between the lines (vectors) (KristaKingMath)
Acute angles between the curves (vectors) (KristaKingMath)
Direction cosines and direction angles (vectors) (KristaKingMath)
Scalar Equation of a Line (KristaKingMath)
Scalar Equation of a Plane (KristaKingMath)
Scalar and vector projections (KristaKingMath)
Cross Product (KristaKingMath)
Vector orthogonal to the plane (KristaKingMath)
Volume of the parallelepiped determined by vectors (KristaKingMath)
Volume of the parallelepiped with adjacent edges (KristaKingMath)
Scalar triple product to verify the vectors are coplanar (KristaKingMath)
Vector and parametric equations of the line (KristaKingMath)
Parametric and symmetric equations of the line (KristaKingMath)
Symmetric Equations of a Line (KristaKingMath)
Parallel, intersecting, skew and perpendicular lines (KristaKingMath)
Equation of the plane using vectors (KristaKingMath)
Point of intersection of a line and a plane (KristaKingMath)
Parallel, perpendicular, and angle between planes (KristaKingMath)
Parametric equations for the line of intersection of two planes (KristaKingMath)
Symmetric equations for the line of intersection of two planes (KristaKingMath)
Distance between a point and a line (vectors) (KristaKingMath)
Distance between a point and a plane (vectors) (KristaKingMath)
Distance between parallel planes (vectors) (KristaKingMath)
Sketching the quadric surface (KristaKingMath)
Reducing a quadric surface equation to standard form (KristaKingMath)
Domain of the vector function (KristaKingMath)
Limit of the vector function (KristaKingMath)
Sketching the vector equation (KristaKingMath)
Projections of the curve onto the coordinate axes (KristaKingMath)
Vector and parametric equations of the line segment (KristaKingMath)
Vector function for the curve of intersection of two surfaces (KristaKingMath)
Derivative of the vector function (KristaKingMath)
Unit tangent vector (KristaKingMath)
Parametric equations of the tangent line (vectors) (KristaKingMath)
Integral of the vector function (KristaKingMath)
arc length of a vector function (KristaKingMath)
reparametrizing the curve in terms of arc length (KristaKingMath)
unit tangent and unit normal vectors (KristaKingMath)
curvature of the vector function (KristaKingMath)
maximum curvature of the function (KristaKingMath)
normal and osculating planes (KristaKingMath)
Velocity & Acceleration Vectors (KristaKingMath)
velocity, acceleration and speed, given position (KristaKingMath)
velocity and position given acceleration and initial conditions (KristaKingMath)
tangential and normal components of the acceleration vector (KristaKingMath)
line integral of a curve (KristaKingMath)
line integral of a vector function (KristaKingMath)
potential function of a conservative vector field (KristaKingMath)
potential function of the conservative vector field to evaluate a line integral (KristaKingMath)
independence of path (KristaKingMath)
work done by the force field (KristaKingMath)
Green's Theorem One Region (KristaKingMath)
Green's Theorem Two Regions (KristaKingMath)
curl and divergence (KristaKingMath)
potential function of the conservative vector field, three dimensions (KristaKingMath)
points on the surface (KristaKingMath)
surface of the vector equation (KristaKingMath)
parametric representation of the surface (KristaKingMath)
tangent plane to the parametric surface (KristaKingMath)
area of the surface (KristaKingMath)
surface integral (KristaKingMath)
surface integral, example 2 (KristaKingMath)
Sketching direction fields (KristaKingMath)
Linear Differential Equations (KristaKingMath)
Circuits and linear differential equations (KristaKingMath)
Linear differential equation initial value problem (KristaKingMath)
Differential Equations (KristaKingMath)
Differential Equations Example 2 (KristaKingMath)
Change of variable to solve a differential equations (KristaKingMath)
Separable differential equations initial value problem (KristaKingMath)
Mixing problems with separable differential equations (KristaKingMath)
exact differential equations (KristaKingMath)
Population Growth (KristaKingMath)
Logistic growth model of a population (KristaKingMath)
Predator-prey systems (KristaKingMath)
Euler's Method Example 1 PART 1/3 (KristaKingMath)
Euler's Method Example 1 PART 2/3 (KristaKingMath)
Euler's Method Example 1 PART 3/3 (KristaKingMath)
Euler's Method Example 2 PART 1/2 (KristaKingMath)
Euler's Method Example 2 PART 2/2 (KristaKingMath)
undetermined coefficients, example 3 (KristaKingMath)
undetermined coefficients, example 4 (KristaKingMath)
Complex conjugate roots of second-order homogeneous differential equations (KristaKingMath)
Second-Order Differential Equations Initial Value Problems Example 1 (KristaKingMath)
Second-Order Differential Equations Initial Value Problems Example 2 (KristaKingMath)
Second-Order Differential Equations Initial Value Problems Example 3 (KristaKingMath)
Second-Order Differential Equations Initial Value Problems Example 4 (KristaKingMath)
Boundary value problem, second-order homogeneous differential equation, distinct real roots
Boundary value problem, second-order homogeneous differential equation, complex conjugate roots
Second-Order Differential Equations - Working Backwards (KristaKingMath)
Second-Order Non-Homogeneous Differential (KristaKingMath)
Second-Order Non-Homogeneous Differential Equations 2 (KristaKingMath)
Second-Order Non-Homogeneous Differential Equation Initial Value Problem (KristaKingMath)
Variation of Parameters for Differential Equations (KristaKingMath)
Laplace Transforms Using the Definition (KristaKingMath)
Laplace Transforms Using a Table (KristaKingMath)
Initial Value Problems with Laplace Transforms (KristaKingMath)
Laplace Transforms and Integration by Parts with Three Functions (KristaKingMath)
Inverse Laplace Transform (KristaKingMath)
Convolution Integral for Initial Value Problems (KristaKingMath)
How to solve EXACT DIFFERENTIAL EQUATIONS IVPs
Partial derivatives - How to solve?
What does a double integral represent?
What does a triple integral represent?
How do you sketch level curves of multivariable functions?