Multivariable Calculus

Vectors and Dot Products

01Vectors, Components, and Length in Space 02Scaling and Adding Vectors: The Parallelogram Rule 03The Dot Product and the Law of Cosines 04Finding Angles in Space With the Dot Product 05The Perpendicularity Test and the Normal Vector to a Plane Problem set0/10 Problem set 20/10 MIT problem set0/2

Determinants and the Cross Product

01The Component of a Vector Along Any Unit Direction 02The 2x2 Determinant and the Area of a Parallelogram 03The 3x3 Determinant and the Volume of a Box 04The Cross Product and the Right-Hand Rule 05The Triple Product and the Volume of a Parallelepiped Problem set0/10 Problem set 20/10 MIT problem set0/2

Matrices and Inverse Matrices

01Why Swapping a Cross Product Flips Its Sign 02The Equation of a Plane Through Three Points 03Matrix Multiplication: Rows Dotted With Columns 04Matrices as Transformations: Identity and Rotation 05The Inverse Matrix and Solving AX = B 06Computing a 3x3 Inverse: Minors, Cofactors, Adjoint Problem set0/10 Problem set 20/10 MIT problem set0/2

Planes and Square Systems

01Planes From a Point and a Normal Vector 02Reading the Normal Vector Off a Plane Equation 03Three Planes, One Point: Solving a 3x3 System 04Invertibility, det A, and the Homogeneous System 05When det A = 0: Coplanar Normals and a Line of Solutions Problem set0/10 Problem set 20/10

Parametric Lines and Curves

01Parametric Equations of a Line Through Two Points 02Where a Line Meets a Plane, and When It Never Does 03The Cycloid: Choosing a Parameter That Is Not Time 04Deriving the Cycloid's Equations From Three Vectors 05Is the Cycloid's Bottom a Corner? Taylor Expansion Finds a Cusp Problem set0/10 Problem set 20/10 MIT problem set0/2

Velocity and Kepler's Law

01Velocity and Speed on the Cycloid: Why the Point Stops Dead 02Acceleration Is a Vector: Turning at Constant Speed Still Counts 03Arc Length and the Unit Tangent: v = T̂ ds/dt 04Equal Areas in Equal Times: Kepler's Second Law and |r × v|05Kepler's Second Law Means the Force Is Central Problem set0/10 Problem set 20/10 MIT problem set0/1

Functions of Several Variables

01Functions of Two Variables: Graphs as Surfaces, Built from Slices 02Level Curves and Contour Plots: A Surface Drawn as a Flat Map 03Reading Contour Plots: Domes, Bowls, Saddles and Mountain Passes 04Partial Derivatives: Change One Variable, Hold the Other Fixed Problem set0/10 Problem set 20/10 MIT problem set0/2

Optimization and Least Squares

01Approximation Formula: Changes in x and y Add, via the Tangent Plane 02Critical Points: Where Both Partial Derivatives Are Zero 03Minimum, Maximum or Saddle Point? Classifying by Completing the Square 04Least Squares: The Best-Fit Line as a Minimization in a and b 05Solving Least Squares: Best-Fit Lines, Exponentials and Parabolas Problem set0/10 Problem set 20/10 MIT problem set0/3

The Second Derivative Test

01Quadratic Critical Points: The Sign of 4ac − b² Decides 02The Second Derivative Test: Classifying by the Sign of AC − B² 03Global Min and Max of x + y + 1/(xy): Boundary and Infinity Problem set0/10 Problem set 20/10 MIT problem set0/1

Differentials and the Chain Rule

01The Total Differential: Why df Is Not Δf, and Its Three Uses 02The Chain Rule: Why It Holds, a Worked Check, and the Product Rule 03Chain Rule for a Change of Variables: Partials in Polar Coordinates Problem set0/10 Problem set 20/10 MIT problem set0/1

Gradients and Constraints

01The Gradient Vector: Why It Is Perpendicular to Level Sets 02Tangent Plane to a Surface From the Gradient: The Hyperboloid 03Directional Derivatives and the Direction of Steepest Ascent Problem set0/10 Problem set 20/10 MIT problem set0/2

01Lagrange Multipliers: Tangent Level Curves and ∇f = λ∇g 02Solving the Lagrange Equations, and Why the Method Works 03The Least-Area Gold Pyramid: Its Top Sits Over the Incenter Problem set0/10 Problem set 20/10 MIT problem set0/1

01Implicit Partial Derivatives: Differentiate the Constraint 02Which Variable Is Held Constant? Subscript Notation 03A Constrained Rate Three Ways: Differentials vs Chain Rule Problem set0/10 Problem set 20/10 MIT problem set0/3

Multiple Integrals

01The Heat Equation: A First Partial Differential Equation 02Double Integrals: Volume Under a Surface by Slicing 03Iterated Integrals Over a Square and a Quarter Disk 04Exchanging the Order of Integration: Redraw the Region Problem set0/10 Problem set 20/10 MIT problem set0/1

01Double Integrals in Polar Coordinates: Why dA = r dr dθ 02Area, Mass, Averages and Center of Mass by Double Integrals 03Moment of Inertia: Spinning a Disk About Its Center and Its Rim Problem set0/10 Problem set 20/10 MIT problem set0/3

Lagrange Multipliers: Tangent Level Curves and ∇f = λ∇g

Why must the gradient of f line up with the gradient of the constraint at the closest point on a hyperbola?

Loading…

←Previous Directional Derivatives and the Direction of Steepest Ascent Next Solving the Lagrange Equations, and Why the Method Works →

Your summary note

1. Optimizing $f$ subject to a constraint $g = c$
  State the constrained problem, record that critical points of $f$ generally fail the constraint, and note that solving $g = c$ for one variable and substituting works only for simple constraints.
2. Tangency at the closest point of $xy = 3$
  Draw the circles $x^2+y^2=k$ over the hyperbola $xy=3$ , shrink $k$ until the circle barely touches it, and record that the minimum occurs where the level curve of $f$ is tangent to $g=3$ .
3. Parallel gradients and the multiplier $\lambda$
  Write the step from tangent level curves to $\nabla f \parallel \nabla g$ through normal vectors, define $\lambda$ by $\nabla f=\lambda \nabla g$ , and record that $\lambda$ may be positive or negative.
4. The Lagrange system
  Write $f_x=\lambda g_x$ , $f_y=\lambda g_y$ , $g=c$ as three equations in $x$ , $y$ , $\lambda$ , and set them up for the hyperbola as $2x=\lambda y$ , $2y=\lambda x$ , $xy=3$ .
Attempt 1 of 2