Calculus III - Multivariable Calculus - Vectors, Curves, Partial Derivatives, Multiple Integrals, Optimization

Embark on a comprehensive journey through multivariable calculus with this full course playlist, covering essential topics from vectors and curves to partial derivatives and multiple integrals. Develop a strong geometric and conceptual understanding of calculus in three dimensions, following the MATH200 curriculum taught by Dr. Trefor Bazett at the University of Victoria. Explore vector operations, lines, and planes before delving into vector-valued functions, learning to plot curves, calculate tangents, curvature, and arclength. Master partial derivatives and their applications in optimization problems, including the use of Lagrange multipliers. Conclude with an in-depth study of integration techniques, from double and triple integrals to polar and spherical coordinates, and the change of variables theorem using the Jacobian. Throughout the 5-hour course, engage with clear explanations and practical examples that illuminate the big ideas of multivariable calculus, preparing you for advanced mathematical concepts and real-world applications.

What are the big ideas of Multivariable Calculus?? Full Course Intro.
Angle between vectors leads to defining the Dot Product | Multivariable Calculus.
Geometrically Defining the Cross Product | Multivariable Calculus.
The Vector Equation of Lines | Multivariable Calculus.
Equations of Planes: Vector & Component Forms | Multivariable Calculus.
3D Curves and their Tangents | Intro to Vector-Valued Functions.
How long is a curve?? The Arclength Formula in 3D.
How curvy is a curve? Intro to Curvature & Circles of Curvature | Multi-variable Calculus.
Torsion: How curves twist in space, and the TNB or Frenet Frame.
Tangential and Normal components of Acceleration | Multi-variable Calculus.
Visualizing Multi-variable Functions with Contour Plots.
Limits are...weird...for multi-variable functions | Limits along paths.
Computing Multivariable Limits Algebraically.
What are derivatives in 3D? Intro to Partial Derivatives.
Continuity vs Partial Derivatives vs Differentiability | My Favorite Multivariable Function.
What is differentiability for multivariable functions??.
The Multi-Variable Chain Rule: Derivatives of Compositions.
Directional Derivatives | What's the slope in any direction?.
Geometric Meaning of the Gradient Vector.
How to find the TANGENT PLANE | Linear approximation of multi-variable functions.
Multi-variable Optimization & the Second Derivative Test.
Multivariable Optimization with Boundaries.
Lagrange Multipliers | Geometric Meaning & Full Example.
Lagrange Multipliers with TWO constraints | Multivariable Optimization.
Defining Double Integration with Riemann Sums | Volume under a Surface.
Double Integration Example over General Regions --- two ways!.
Change the order of integration to solve tricky integrals.
Double Integration in Polar Coordinates | Example & Derivation.
The Gaussian Integral // Solved Using Polar Coordinates.
Triple Integrals in Cartesian Coordinates | Volume between Surfaces.
Integration in Spherical Coordinates.
Change of Variables & The Jacobian | Multi-variable Integration.