Multivariable Calculus (MIT OCW)

schedule

week 1

• Part A: Vectors, Determinants, Planes
• Part B: Matrices and Systems of Equations
• Part C: Parametric Equations for Curves

week 2

• Part A: Functions of Two Variables, Tangent Approximation and Optimization
• Part B: Chain Rule, Gradient and Directional Derivatives
• Part C: Lagrange Multipliers and Constrained Differentials

week 3

• Part A: Double Integrals
• Part B: Vector Fields and Line Integrals
• Part C: Green’s Theorem

week 4

• Part A: Triple Integrals
• Part B: Flux and the Divergence Theorem
• Part C: Line Integrals and Stokes’ Theorem

Vectors and Matrices

Part A: Vectors, Determinants, Planes

• vector: has direction and magnitude/length
  • represented using coordinate system

a₁î + a₂ĵ + a₃k̂ =  < a₁, a₂, a₃> $$|\vec{A}| = \sqrt{a_1 + a_2 + a_3}$$

A⃗ ⋅ B⃗ = ∑a_ib_i = a₁b₁ + a₂b₂ + a₃b₃

geometrically A⃗ ⋅ B⃗ = |A⃗||B⃗|cosθ

• law of cosines: if C⃗ = A⃗ − B⃗, then define

|C⃗| = |A⃗|² + |B⃗|² − 2|A⃗||B⃗|cosθ

we know |C⃗|² = C⃗ ⋅ C⃗ = (A⃗ − B⃗) ⋅ (A⃗ − B⃗) = |A⃗|² + |B⃗|² − 2A⃗ ⋅ B⃗

then we get: 2|A⃗||B⃗|cosθ = 2A⃗ ⋅ B⃗

• the dot product can be used for
  • computing lengths and angles
  • detect orthogonality
    • A⃗ ⋅ B⃗ = 0 means they are perpendicular to each other, cosθ, θ = 0^deg, 90^deg
  • vector components

1. vector components

   • find a component A⃗ along direction u⃗, where u⃗ is a unit vector: |A⃗cosθ|
     • orthogonal projection of A on u⃗, leg parallel to u⃗: |A|cosθ or $$\frac{\vec{A} \vec{u}}{|\vec{u}|}$$
   • component (length) of a vector is scalar

2. area and determinants in 2D

   determinant of A⃗, B⃗ is +/- area of a parallelogram, |A⃗||B⃗|sinθ = det(A⃗, B⃗)

   area of triangle: $$\frac{1}{2} det(\vec{A}, \vec{B})$$

3. volumes and determinants in space

   • det(A⃗, B⃗, C⃗) = +/- volume of parallelepiped
   • A_ij = (−1)^i + j|A_ij|

4. cross products

   • 2 vectors in 3D space => vector

   $$\vec{A} \times \vec{B} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k}\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\\ \end{vmatrix}$$

   • theorem:
     • the length of the cross product of A⃗ and B⃗, is the area of the parallelogram
     • the direction of the cross product of A⃗ and B⃗ is perpendicular to the plane of the parallelogram
       • direction is based on the right-hand rule
   • volume of parallelepiped:
     • area(base) times height
     • |B⃗ × C⃗| × (A⃗ ⋅ n⃗)
       • where n⃗ is the vector perpendicular to $$\frac{\vec{B} \times \vec{C}}{|\vec{B} \times \vec{C}|}$$
   • rules of cross products
     • B⃗ × C⃗ ≠ C⃗ × B⃗, B⃗ × C⃗ = −C⃗ × B⃗
     • B⃗ × B⃗ = 0

5. equations of planes

   • application of cross product: 3 points in space, equation of plane P₁, P₂, P₃, is point P on that plane?
     • consider vectors P₁, P, P₁, P₂, P₁, P₃: is the parallelogram completely flat?
       • is det(P₁P, P₁P₂, P₁P₃) = 0
     • P is in the plane, only when $$\vec{P_1 P}$$ is perpendicular N⃗, where N⃗ is a cross product
       • $$\vec{P_1 P} \cdot (\vec{P_1 P_2} \times \vec{P_1 P_3}) = 0$$

Part B: Matrices and Systems of Equations

1. matrix multiplication

   • often have linear relations
   • entries in matrix product AX
     • dot products between rows of A and columns of X
   • product AB represents doing transformation B then transformation A
   • identity matrix: do nothing

2. matrix inverse

   • if we have function of u in terms of x, can use matrix inverse to express function of x in terms of u

   • if A is the inverse of M: AM = I, MA = I

     • M = A⁻¹

   • if AX = B

   • A⁻¹(AX) = A⁻¹B

   • X = A⁻¹B

   • invert matrix by hand:

     • $$A^{-1} = \frac{1}{det(A)}adj(A)$$

3. equations of planes ii

   ax + by + cz = d

   • find plane perpendicular to a vector
     • point P is in the plane, whenever $$\vec{OP} \cdot \vec{N} = 0$$
     • if plane goes through origin
     •  < a, b, c>, is the normal vector for the plane
   • plane in point-normal form:
     • need point P₀ and vector N orthogonal to plane, we can also say N is normal to the plane
       • since the vector $$\vec{P_0 P}$$ is in the plane, we can say $$N \cdot \vec{P_0 P} = 0$$

   1. distance to plane/line

      1. point to plane: need a point P, plane with normal N⃗ and containing point Q $$\text{distance} = |PR| = |\vec{PQ}| \text{cos} \theta = |\vec{PQ} \cdot \frac{N}{|N|}|$$
      2. point to line: point P, line with direction vector v and containing point Q $$d = |QP| \text{sin} \theta = |\vec{QP} \cdot \frac{v}{|v|}|$$
      3. distance between parallel planes
         • reduce the problem to distance from point to plane
      4. distance between skew lines
         • place lines in parallel planes and find distance between planes
         • find normal M = v₁ × v₂

4. linear systems and planes

   • 3 planes intersect => two planes intersect and form a line => find the line that intersects in the third plane
   • if set of solutions to 3by3 linear system is not a single point => no solution, line or plane
     • 3rd plane might be parallel to the line where the first two planes intersect
       • if line contained in third plane: infinite solution
       • if line is parallel to third plane and not contained, then there are no solutions

5. solutions to square systems

   • can’t always find the inverse of the matrix; when the determinant is 0
   • homogenous system: right hand side is 0
     • origin is always a solution
       • if det(A) is not 0: can solve the system using inverse of A
       • if det(A) = 0: det(vecN₁, vecN₂, vecN₃) = 0 then vecN₁, vecN₂, vecN₃ are coplanar (in the same plane)
         • infinite many solutions
   • general case:
     • if det(A) is not zero: X = A⁻¹B
     • if det(A) is zero, either no solution or infinitely many solutions

Part C: Parametric Equations for Curves

1. Session 15: Equations of Lines

   • intersection of 2 planes
   • trajectory of moving point
   • paramatric form: point moving in space traces out path over time => position of point and time
   • parametric form: equation of line that passes through the point P and is parallel to the vector v⃗ r⃗ = r⃗₀ + tv⃗ =  < x₀ + tv₁, y₀ + tv₂, z₀ + tv₃>, where r⃗₀ is the vector using the point P
     • then you break the vector into equations
   • parametric form: intersection of two planes
   • parametric form: line through two points

2. Session 16: Intersection of a Line and a Plane

   • two points vs. a line
     • substitute the point, and see if value is larger than the sum of the unknowns => tells you which side the points are on
   • if line is in the plane, than plugging (x(t), y(t), z(t)), we always get the value for the plane
   • if the line is parallel to the plane, then never get that value, get another constant

3. Session 17: General Parametric Equations; the Cycloid

   • parametric curves:

4. Session 18: Point (Cusp) on Cycloid

5. Session 19: Velocity and Acceleration

6. Session 20: Velocity and Arc Length

7. Session 21: Kepler’s Second Law

Partial Derivatives

Part A: Functions of Two Variables, Tangent Approximation and Optimization

1. Session 24: Functions of Two Variables: Graphs

2. Session 25: Level Curves and Contour Plots

3. Session 26: Partial Derivatives

4. Session 27: Approximation Formula

5. Session 28: Optimization Problems

6. Session 29: Least Squares

7. Session 30: Second Derivative Test

8. Session 31: Example

Part B: Chain Rule, Gradient and Directional Derivatives

1. Session 32: Total Differentials and the Chain Rule

2. Session 33: Examples

3. Session 34: The Chain Rule with More Variables

4. Session 35: Gradient: Definition, Perpendicular to Level Curves

5. Session 36: Proof

6. Session 37: Example

7. Session 38: Directional Derivatives

Part C: Lagrange Multipliers and Constrained Differentials

1. Session 39: Statement of Lagrange Multipliers and Example

2. Session 40: Proof of Lagrange Multipliers

3. Session 41: Advanced Example

4. Session 42: Constrained Differentials

5. Session 43: Clearer Notation

6. Session 44: Example

Double Integrals and Line Integrals in the Plane

Part A: Double Integrals

1. Session 47: Definition of Double Integration

2. Session 48: Examples of Double Integration

3. Session 49: Exchanging the Order of Integration

4. Session 50: Double Integrals in Polar Coordinates

5. Session 51: Applications: Mass and Average Value

6. Session 52: Applications: Moment of Inertia

7. Session 53: Change of Variables

8. Session 54: Example: Polar Coordinates

9. Session 55: Example

Part B: Vector Fields and Line Integrals

1. Session 56: Vector Fields

2. Session 57: Work and Line Integrals

3. Session 58: Geometric Approach

4. Session 59: Example: Line Integrals for Work

5. Session 60: Fundamental Theorem for Line Integrals

6. Session 61: Conservative Fields, Path Independence, Exact Differentials

7. Session 62: Gradient Fields

8. Session 63: Potential Functions

9. Session 64: Curl

Part C: Green’s Theorem

1. Session 65: Green’s Theorem

2. Session 66: Curl(F) = 0 Implies Conservative

3. Session 67: Proof of Green’s Theorem

4. Session 68: Planimeter: Green’s Theorem and Area

5. Session 69: Flux in 2D

6. Session 70: Normal Form of Green’s Theorem

7. Session 71: Extended Green’s Theorem: Boundaries with Multiple Pieces

8. Session 72: Simply Connected Regions and Conservative

Triple Integrals and Surface Integrals in 3-Space

Part A: Triple Integrals

1. Session 74: Triple Integrals: Rectangular and Cylindrical Coordinates

2. Session 75: Applications and Examples

3. Session 76: Spherical Coordinates

4. Session 77: Triple Integrals in Spherical Coordinates

5. Session 78: Applications: Gravitational Attraction

Part B: Flux and the Divergence Theorem

1. Session 79: Vector Fields in Space

2. Session 80: Flux Through a Surface

3. Session 81: Calculating Flux; Finding ndS

4. Session 82: ndS for a Surface z = f(x, y)

5. Session 83: Other Ways to Find ndS

6. Session 84: Divergence Theorem

7. Session 85: Physical Meaning of Flux; Del Notation

8. Session 86: Proof of the Divergence Theorem

9. Session 87: Diffusion Equation

Part C: Line Integrals and Stokes’ Theorem

1. Session 88: Line Integrals in Space

2. Session 89: Gradient Fields and Potential Functions

3. Session 90: Curl in 3D

4. Session 91: Stokes’ Theorem

5. Session 92: Proof of Stokes’ Theorem

6. Session 93: Example

7. Session 94: Simply Connected Regions; Topology

8. Session 95: Stokes’ Theorem and Surface Independence

9. Session 96: Summary of Multiple Integration