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Multivariable Calculus

49 cards · 🇺🇸 English

Partial derivatives, multiple integrals and vector calculus basics.

calculusmultivariable

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Multivariable calculus

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  1. 1Multivariable calculus
    The extension of calculus to functions of several variables, involving differentiation and integration of multivariate functions
  2. 2Another name for multivariable calculus
    Multivariate calculus
  3. 3Vector calculus
    The special case of multivariable calculus in three-dimensional space
  4. 4Key difference between 1D and higher-dimensional spaces for limits
    In higher dimensions, there are infinite ways to approach a point (versus only two directions in 1D)
  5. 5What consequence does multiple approach paths have?
    Limits and continuity must be defined more carefully; directional limits may differ
  6. 6Limit along a path
    The limit of f(γ(t)) as t → a, where γ(t) is a parametrized path in n-dimensional space
  7. 7Does the limit along a path depend on the path chosen?
    Yes, the limit value can depend on the form of the path, not just the point approached
  8. 8Multivariate continuity
    f is continuous at p if lim(t→a) f(γ(t)) = f(p) for all continuous paths γ with γ(a) = p
  9. 9Is continuity along one path sufficient for multivariate continuity?
    No, being continuous along one path does not imply multivariate continuity
  10. 10Is continuity in each argument sufficient for multivariate continuity?
    No, a function can be continuous in x for fixed y and continuous in y for fixed x but not multivariate continuous
  11. 11Directional derivative
    The derivative of a scalar-valued function along a unit vector v at point p: D_v f(p) = lim(h→0) [f(p + hv) - f(p)]/h
  12. 12Does the directional derivative depend on the entire path?
    No, it depends only on the tangent vector of the path at the point (if f is Lipschitz continuous)
  13. 13Can a unique scalar derivative be defined without specifying a direction?
    No, it's not possible to define a unique scalar derivative without a direction
  14. 14Partial derivative
    A derivative with respect to one variable with all other variables held constant
  15. 15How can a partial derivative be viewed?
    As the directional derivative of the function along a coordinate axis
  16. 16Del operator (∇)
    An operator used to define gradient, divergence, and curl in terms of partial derivatives
  17. 17Jacobian matrix
    A matrix of partial derivatives representing the derivative of a function between spaces of arbitrary dimension
  18. 18How can the derivative be understood in multivariable calculus?
    As a linear transformation which varies from point to point in the domain of the function
  19. 19Gradient
    A vector of all partial derivatives: ∇f = (∂f/∂x₁, ∂f/∂x₂, ..., ∂f/∂xₙ)
  20. 20Divergence
    The dot product of del with a vector field: div F = ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z
  21. 21Curl
    The cross product of del with a vector field: curl F = ∇×F
  22. 22Multiple integral
    An extension of the integral to functions of any number of variables (e.g., double, triple integrals)
  23. 23Fubini's theorem
    A multiple integral may be evaluated as a repeated/iterated integral if the integrand is continuous
  24. 24Surface integral
    An integral used to integrate over curved surfaces (2D manifolds in 3D space)
  25. 25Line integral
    An integral used to integrate over curves (1D manifolds)
  26. 26Gradient theorem
    ∫_C ∇f·dr = f(b) - f(a), relating line integrals of gradients to function values at endpoints
  27. 27Stokes' theorem
    ∫∫_S (∇×F)·dS = ∫_C F·dr, relating surface integrals of curl to line integrals around boundary
  28. 28Divergence theorem
    ∫∫∫_V (∇·F) dV = ∫∫_S F·dS, relating volume integrals of divergence to surface integrals (also called Gauss's theorem)
  29. 29Green's theorem
    ∫∫_D (∂Q/∂x - ∂P/∂y) dA = ∫_C P dx + Q dy, a 2D version of Stokes' theorem
  30. 30Generalized Stokes' theorem
    A unified theorem that applies to integration of differential forms over manifolds (includes all four integral theorems)

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