Pythagorean Theorem

Examples of Pythagorean Theorem in the following topics:

• Arc Length and Speed

  • Arc length and speed in parametric equations can be calculated using integration and the Pythagorean theorem.
  • This equation is obtained using the Pythagorean Theorem.
  • By the Pythagorean Theorem, each hypotenuse will have length $\sqrt{dx^2 + dy^2}$.
  • As shown previously using the Pythagorean Theorem, it is given by:

• Arc Length and Speed

  • To find this using integration, we should start out by using the Pythagorean Theorem for length of the different sides of a triangle:
  • For a small piece of curve, ∆s can be approximated with the Pythagorean theorem.

• Arc Length and Surface Area

  • According to Pythagoras's theorem $ds^2=dx^2+dy^2$, from which:
  • For a small piece of curve, $\Delta s$ can be approximated with the Pythagorean theorem.

• Area of a Surface of Revolution

  • The quantity $\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2$ comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula.

• Area and Distances

  • Integration is connected with differentiation through the fundamental theorem of calculus: if $f$ is a continuous real-valued function defined on a closed interval $[a,b]$, then, once an antiderivative F of f is known, the definite integral of $f$ over that interval is given by$\int_{a}^{b}f(x)dx = F(b) - F(a)$.
  • Applying the fundamental theorem of calculus to the square root curve, $f(x) = x^{1/2}$, we look at the antiderivative, $F(x) = \frac{2}{3} \cdot x^\frac{3}{2}$, and simply take $F(1) − F(0)$, where $0$ and $1$ are the boundaries of the interval $[0,1]$.
  • According to Pythagoras's theorem, $ds^2=dx^2+dy^2$ , from which we can determine:
  • For a small piece of curve, $\Delta s$ can be approximated with the Pythagorean 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

• 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