Fubini's theorem

Examples of Fubini's theorem in the following topics:

• Iterated Integrals

  • A theorem called Fubini's theorem, however, states that they may be equal under very mild conditions.

• Double Integrals Over Rectangles

  • Here, we exchanged the order of the integration, assuming that $f(x,y)$ satisfies the conditions to apply Fubini's theorem.

• Using the Yield Curve to Estimate Interest Rates in the Future

  • The equation itself is a rather evolved derivation, incorporating bond prices, forward rates, risk free rates, the Wiener process, Leibniz's rule, and Fubini's Theorem.

• Trigonometry and Complex Numbers: De Moivre's Theorem

• Stokes' Theorem

  • Stokes' theorem relates the integral of the curl of a vector field over a surface to the line integral of the field around the boundary.
  • The generalized Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
  • The Kelvin–Stokes theorem, also known as the curl theorem, is a theorem in vector calculus on $R^3$.
  • The Kelvin–Stokes theorem is a special case of the "generalized Stokes' theorem."
  • As we have seen in our previous atom on gradient theorem, this simply means:

• Student Learning Outcomes

• Green's Theorem

  • Green's theorem gives relationship between a line integral around closed curve $C$ and a double integral over plane region $D$ bounded by $C$.
  • Green's theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the $xy$-plane.
  • Considering only two-dimensional vector fields, Green's theorem is equivalent to the two-dimensional version of the divergence theorem.
  • Green's theorem can be used to compute area by line integral.
  • Explain the relationship between the Green's theorem, the Kelvin–Stokes theorem, and the divergence theorem

• The Fundamental Theorem of Calculus

  • The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function to the concept of the integral.
  • The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function to the concept of the integral.
  • There are two parts to the theorem.
  • The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by differentiation.
  • We can see from this picture that the Fundamental Theorem of Calculus works.

• The Divergence Theorem

  • The divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.
  • In physics and engineering, the divergence theorem is usually applied in three dimensions.
  • In one dimension, it is equivalent to the fundamental theorem of calculus.
  • The theorem is a special case of the generalized Stokes' theorem.
  • Apply the divergence theorem to evaluate the outward flux of a vector field through a closed surface

• The Mean Value Theorem, Rolle's Theorem, and Monotonicity

  • In calculus, the mean value theorem states, roughly: given a planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints .
  • The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval.
  • This theorem can be understood intuitively by applying it to motion: If a car travels one hundred miles in one hour, then its average speed during that time was 100 miles per hour.
  • The mean value theorem follows from the more specific statement of Rolle's theorem, and can be used to prove the more general statement of Taylor's theorem (with Lagrange form of the remainder term).
  • Use the Mean Value Theorem and Rolle's Theorem to reach conclusions about points on continuous and differentiable functions