Math Insight

Thread: Math 2374

Topics as covered in the Math 2374 course at the University of Minnesota. Pages marked with an asterisk * are the pre-lecture pages that students are required to read before lecture.

• Part 0
  • An introduction to vectors
  • Vectors in two- and three-dimensional Cartesian coordinates
  • The dot product
  • The formula for the dot product in terms of vector components
  • Dot product examples
• Part 1
  • Math 2374 introduction to Math Insight
  • Parametrization of a line
  • Parametrization of a line examples
  • Matrices and determinants
  • The cross product
  • The formula for the cross product
  • Cross product examples
  • The scalar triple product
  • Scalar triple product example
  • The relationship between determinants and area or volume
• Part 2
  • Forming planes*
  • Forming plane examples
  • A line or a plane or a point?
  • Distance from point to plane
  • Distance from point to plane example
  • Intersecting planes example
  • Parametrization of a plane
  • Plane parametrization example
• Part 3
  • Vectors in arbitrary dimensions
  • Examples of n-dimensional vectors
  • Introduction to matrices
  • Multiplying matrices and vectors*
  • Matrix and vector multiplication examples
  • Dot product in matrix notation
  • Matrices and linear transformations
  • Determinants and linear transformations
  • Geometric properties of the determinant
  • How linear transformations map parallelograms and parallelepipeds
• Part 4
  • Level sets*
  • Level set examples
  • Function notation
  • Translation, rescaling, and reflection
• Part 5
  • Introduction to partial derivatives*
  • Partial derivative examples
  • Partial derivative by limit definition
• Part 6
  • Introduction to differentiability*
  • The derivative matrix
  • The linear approximation
  • Examples of calculating the derivative
  • The definition of differentiabililty
  • Subtleties of differentiability
  • The multidimensional differentiability theorem
  • Non-differentiable functions must have discontinuous partial derivatives
  • A differentiable function with discontinuous partial derivatives
• Part 7
  • Introduction to the chain rule*
  • Special cases of the chain rule
  • Chain rule examples
  • An introduction to parametrized curves
  • Derivatives of parameterized curves
  • Parametrized curve and derivative as location and velocity
  • Tangent lines to parametrized curves
  • Tangent line to parametrized curve examples
  • Orienting curves
• Part 8
  • An introduction to the directional derivative and the gradient*
  • The gradient vector
  • Derivation of the directional derivative and the gradient
  • Directional derivative and gradient examples
• Part 9
  • Introduction to double integrals*
  • Double integrals as iterated integrals
  • Double integral examples
  • Double integrals as volume
  • Double integrals as area
  • Examples of changing the order of integration in double integrals
  • Double integrals where one integration order is easier
• Part 10
  • Introduction to triple integrals*
  • The shadow method for determining triple integral bounds
  • The cross section method for determining triple integral bounds
  • Triple integral examples
• Part 11
  • The length of a curve*
  • Parametrized curve arc length examples
• Part 12
  • Vector field overview
  • Vector fields as fluid flow
  • The idea of divergence of a vector field*
  • The idea of curl of a vector field*
  • Divergence and curl notation
  • Divergence and curl example
  • Subtleties about divergence
  • Subtleties about curl
  • The components of the curl
• Part 13
  • Introduction to a line integral of a scalar-valued function*
  • Line integrals are independent of parametrization*
  • Examples of scalar line integrals
• Part 14
  • Introduction to a line integral of a vector field
  • Alternate notation for vector line integrals
  • Vector line integral examples
  • Line integrals as circulation
  • The definition of curl from line integrals
  • Calculating the formula for circulation per unit area
• Part 15
  • The idea behind Green's theorem*
  • When Green's theorem applies*
  • Other ways of writing Green's theorem
  • Green's theorem examples
  • Using Green's theorem to find area
  • Green's theorem with multiple boundary components
• Part 16
  • An introduction to conservative vector fields*
  • The gradient theorem for line integrals
  • A simple example of using the gradient theorem
  • How to determine if a vector field is conservative
  • A conservative vector field has no circulation
  • A path-dependent vector field with zero curl
  • Testing if three-dimensional vector fields are conservative
  • Finding a potential function for conservative vector fields
  • Finding a potential function for three-dimensional conservative vector fields
• Part 17
  • Introduction to changing variables in double integrals*
  • Area calculation for changing variables in double integrals
  • Double integral change of variable examples
  • Illustrated example of changing variables in double integrals
• Part 18
  • Triple integral change of variables story*
  • Cylindrical coordinates
  • Spherical coordinates
  • Introduction to changing variables in triple integrals
  • Volume calculation for changing variables in triple integrals
  • Triple integral change of variable examples
• Part 19
  • An introduction to parametrized surfaces*
  • Parametrized surface examples
  • Parametrization of a plane
  • Plane parametrization example
  • Quadric surfaces
• Part 20
  • Surface area of parametrized surfaces*
  • Calculation of the surface area of a parametrized surface
  • Parametrized surface area example
  • Normal vector of parametrized surfaces
  • Orienting surfaces
  • A Möbius strip is not orientable
• Part 21
  • Introduction to a surface integral of a scalar-valued function*
  • Scalar surface integral examples
  • Introduction to a surface integral of a vector field*
  • Vector surface integral examples
• Part 22
  • The idea behind Stokes' theorem*
  • Proper orientation for Stokes' theorem*
  • Stokes' theorem examples
• Part 23
  • The idea behind the divergence theorem*
  • Divergence theorem examples
• Part 24
  • Taylor's theorem for multivariable functions*
  • Multivariable Taylor polynomial example
• Part 25
  • Introduction to local extrema*
  • Two variable local extrema examples
• Part 26
  • The integrals of multivariable calculus
  • The fundamental theorems of vector calculus
  • Length, area, and volume factors

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