Math 20E Fall 2023
Terminology and Theorems

Updated 10/26/23

• Section 1.1 (review)
  • vector addition; scalar multiplication
  • geometry of vector operations
  • coordinates of a point; components of a vector
  • the standard basis vectors
  • point-direction form of a line
  • parametric equation of a line: point-point form
• Section 1.2 (review)
  • the inner product
  • unit vectors
  • length; distance
  • angle between two vectors
  • Cauchy-Schwarz inequality
  • orthogonal projection
  • triangle inequality
  • displacement; velocity
• Section 1.3 (review)
  • matrices (2x2 and 3x3)
  • properties of determinants
  • cross products
  • geometry of 2x2 determinants; geometry of 3x3 determinants
  • equation of a plane in space
  • distance from a point to a plane
• Section 1.4 (review)
  • cylindrical coordinates
  • spherical coordinates
• Section 2.1 (review)
  • functions; mappings
  • graphs of functions
  • level sets, curves and surfaces
  • sections
• Section 2.2 (review)
  • open sets
  • boundary; boundary points
  • limit
  • properties of limits
  • continuity; continuous functions
• Section 2.4 (review)
  • path
  • curve
  • parametrize
  • velocity
  • speed
  • tangent vector
  • tangent line
• Section 2.6 (review)
  • gradient
  • directional derivative
  • gradient normal to level surface
  • tangent plane to level surface
• Section 3.1 (review)
  • iterated partial derivatives
  • mixed partial derivatives
  • equality of mixed partial derivatives
• Section 4.1 (review)
  • differentiation rules for paths
    • sum rule
    • scalar multiplication rule
    • dot product rule
    • cross product rule
    • chain rule
  • acceleration
  • Newton's second law
• Section 4.2 (review)
  • arc length
• Section 2.3
  • partial derivatives
  • linear approximation
  • tangent plane
  • derivative; matrix of partial derivatives
• Section 2.5
  • constant multiple rule
  • sum rule
  • product rule
  • quotient rule
  • chain rule
• Section 5.1
  • double integral
  • slice method; Cavalieri's principle
  • iterated integrals
• Section 5.2
  • Fubini's theorem
• Section 5.3
  • x-simple region
  • y-simple region
  • boundary
  • elementary region
• Section 5.4
  • simple region
  • mean-value inequality
  • mean-value theorem
• Section 6.1
  • one-to-one
  • onto
  • one-to-one and onto mapping
• Section 6.2
  • Jacobian determinant
  • change of variables theorem
    • double integrals
    • triple integrals
  • polar coordinates
  • cylindrical coordinates
  • spherical coordinates
• Section 4.3
  • vector field
    • in the plane
    • in space
  • scalar field; component scalar fields
  • gradient vector field
  • flow line; streamline; integral curve
• Section 7.1
  • path integral, integral of f(x,y,z) along the path c
• Section 7.2
  • line integral of F along c
  • differential form
  • reparametrization
    • orientation-preserving
    • orientation-reversing
  • opposite path
  • change of parametrization for path integrals (theorem)
  • line integrals of gradient vector fields (theorem)
  • simple curve
  • parametrization
  • endpoints
  • oriented simple curve, directed simple curve
• Section 7.3
  • parametrized surface, parametrization of a surface
  • differentiable or C¹ surface
  • tangent vectors to parametrized surfaces
  • regular, smooth
  • tangent plane
• Section 7.4
  • surface area
    • of a graph
    • of a surface of revolution

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