Rolle's theorem

Examples of Rolle's theorem in the following topics:

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

  • 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).
  • Rolle's Theorem states that if a real-valued function $f$ is continuous on a closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and f(a) = f(b), then there exists a c in the open interval $(a, b)$ such that $f'(c)=0$.
  • Use the Mean Value Theorem and Rolle's Theorem to reach conclusions about points on continuous and differentiable functions

• Maximum and Minimum Values

  • For example, if a bounded differentiable function $f$ defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle'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:

• 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

• Fundamental Theorem for Line Integrals

  • Gradient theorem says that a line integral through a gradient field can be evaluated from the field values at the endpoints of the curve.
  • The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
  • The gradient theorem implies that line integrals through irrotational vector fields are path-independent.
  • In physics this theorem is one of the ways of defining a "conservative force."
  • The gradient theorem also has an interesting converse: any conservative vector field can be expressed as the gradient of a scalar field.

• Functions of Several Variables

  • We have also studied theorems linking derivatives and integrals of single variable functions.
  • The theorems we learned are gradient theorem, Stokes' theorem, divergence theorem, and Green's theorem.
  • In a more advanced study of multivariable calculus, it is seen that these four theorems are specific incarnations of a more general theorem, the generalized Stokes' theorem, which applies to the integration of differential forms over manifolds.

• Calculating Limits Using the Limit Laws

  • Limits of functions can often be determined using simple laws, such as L'Hôpital's rule and squeeze theorem.
  • In this atom, we will study two examples: L'Hôpital's rule or the squeeze theorem.
  • The squeeze theorem is a technical result that is very important in proofs in calculus and mathematical analysis.
  • The squeeze theorem is formally stated as follows:
  • Calculate a limit using simple laws, such as L'Hôpital's Rule or the squeeze theorem

• Parametric Surfaces and Surface Integrals

  • Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form.
  • We will study surface integral of vector fields and related theorems in the following atoms.
  • An illustration of the Kelvin–Stokes theorem, with surface $\Sigma$, its boundary $\partial$, and the "normal" vector $\mathbf{n}$.