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Calculus 3 Lecture 11.1: An Introduction to Vectors: Discovering Vectors with focus on adding, subtracting, position vectors, unit vectors and magnitude.
Calculus 3 Lecture 11.2: Vectors in 3-D Coordinate System: A study of point relationships and vectors in 3-D. Emphasis on position vectors, magnitude, and unit vectors.
Calculus 3 Lecture 11.3: Using the Dot Product: Explanation of the Dot Product, Finding the angle between two vectors including how the Dot Production show orthogonal vectors, Vector Projection, Work, and Direction angles.
Calculus 3 Lecture 11.4: The Cross Product: Finding the Cross Product of two vectors with Determinants, Using the Cross Product to find Mutually Orthogonal Vectors (with proofs), Torque, Area of a Parallelogram, Volume of a Parallelepiped, and Coplanar Vectors.
Calculus 3 Lecture 11.5: Lines and Planes in 3-D: Parameter and Symmetric Equations of Lines, Intersection of Lines, Equations of Planes, Normals, Relationships between Lines and Planes, and Distances between 3-D objects.
Calculus 3 Lecture 11.6: Cylinders and Surfaces in 3-D: How to Identify and Sketch Cylinders and Surfaces. Shows step by step instruction for any cylinder, ellipsoid, hyperboloid, cone, paraboloid, and hyperbolic paraboloid. Focus is on HAND SKETCHING techniques.
Calculus 3 Lecture 11.7: Using Cylindrical and Spherical Coordinates: Show how to convert between Rectangular, Cylindrical, and Spherical coordinates AND how to convert between Rectangular, Cylindrical, and Spherical Equations.
Calculus 3 Lecture 12.1: An Introduction To Vector Functions: The interpretation of Vector Functions and How to graph Vector Functions. Focus is on the graphing of 3-D Vector Functions
Calculus 3 Lecture 12.2: Derivatives and Integrals of Vector Functions: How to take Derivatives and Integrals of Vector Functions. Focus will be on the Tangent vector to space curves, finding tangent lines to vector functions, and Initial Value problems involving integrals
Calculus 3 Lecture 12.3: Arc Length/Parameterization, TNB (Frenet-Serret) Intro: How to find the arc length and arc length function of a vector function (space curve). How to reparameterize a vector function by arc length. An introduction to Movable Frames; TNB Frames or Frenet-Serret Frames and what they mean.
Calculus 3 Lecture 12.3/12.5: TNB (Frenet-Serret) Frames, Curvature, and Torsion: Focus on how to compute the TNB Frames as simply as possible and use the results to find the Curvature, Osculating circle, osculating plane, normal plane, and radius of curvature. Formulas are also given including how to perform vector encapsulation for 2-D functions.
Calculus 3 Lecture 12.4: Velocity and Acceleration of Vector Functions: How to find the Velocity and Acceleration of a particle when the particular motion is described by a vector function (space curve). How to integrate to find a position function from acceleration or velocity. How to define acceleration as "T" and "N". A look at Projectile Motion.
Calculus 3 Lecture 13.1: Intro to Multivariable Functions (Domain, Sketching, Level Curves): Working with Multivariable Functions with an emphasis on finding Domain, Sketching Domain, Sketching Functions, and finding both Level Curves and Contour Plots.
Calculus 3 Lecture 13.2: Limits and Continuity of Multivariable Functions: How to show a limit exits or Does Not Exist for Multivariable Functions (including Squeeze Theorem). Also, how to determine regions of continuity.
Calculus 3 Lecture 13.3: Partial Derivatives (Derivatives of Multivariable Functions): How to find the slope of a tangent line to a surface in the X-Direction and Y-Direction. Focus is on the concept of a partial derivative and several examples. Also included is implicit differentiation and higher order partial derivatives.
Calculus 3 Lecture 13.4: Finding Differentials of Multivariable Functions: A review of Differentials from Calculus 1 and an extrapolation towards Differentials with more than 1 Independent Variable. Focus will be on the derivation of the idea of Differentials and the application of Differentials in Word Problems.
Calculus 3 Lecture 13.5: The Chain Rule for Multivariable Functions: How to find derivatives of Multivariable Functions involving Parametrics and/or Compositions. Focus will be on deriving the Chain Rule and practice of examples.
Calculus 3 Lecture 13.6: Finding Directional Derivatives and Gradients: How to find a Directional Derivative along the path of any given vector. Also, an introduction to the Gradient and some basics on how it is used.
Calculus 3 Lecture 13.7: Finding Tangent Planes and Normal Lines to Surfaces: How to find a tangent plane and/or a normal line to any surface (multivariable function) at a point. This is based on the ability to find the gradient of a function.
Calculus 3 Lecture 13.8: Finding Extrema of Functions of 2 Variables (Max and Min): How to find Relative (Local) and Absolute Extrema of 2 Variable Functions. Optimization is also discussed.