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The first derivative implied by these parametric equations is = / / = ˙ ˙ (), where the notation ˙ denotes the derivative of x with respect to t. This can be derived using the chain rule for derivatives: d y d t = d y d x ⋅ d x d t {\displaystyle {\frac {dy}{dt}}={\frac {dy}{dx}}\cdot {\frac {dx}{dt}}} and dividing both sides by d x d t ...
In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of ...
The second fundamental form of a parametric surface S in R 3 was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin. Then f and its partial derivatives with respect to x and y vanish at (0,0).
The symmetry may be broken if the function fails to have differentiable partial derivatives, which is possible if Clairaut's theorem is not satisfied (the second partial derivatives are not continuous). The function f(x, y), as shown in equation , does not have symmetric second derivatives at its origin.
The second equation, called the Codazzi equation or Codazzi-Mainardi equation, states that the covariant derivative of the second fundamental form is fully symmetric. It is named for Gaspare Mainardi (1856) and Delfino Codazzi (1868–1869), who independently derived the result, [ 3 ] although it was discovered earlier by Karl Mikhailovich ...
The covariant derivative of a vector field with components is given by: ; = = + and similarly ... Recall that the second fundamental form is, for each ...
[footnote 1] It should not be confused with the normal mathematical script letters P: 𝒫 and 𝓅. In computing, the letter ℘ is available as \wp in TeX. In Unicode the code point is U+2118 ℘ SCRIPT CAPITAL P (℘, ℘), with the more correct alias weierstrass elliptic function. [footnote 2] In HTML, it can be escaped as ℘.
The proof of the general Leibniz rule [2]: 68–69 proceeds by induction. Let f {\displaystyle f} and g {\displaystyle g} be n {\displaystyle n} -times differentiable functions. The base case when n = 1 {\displaystyle n=1} claims that: ( f g ) ′ = f ′ g + f g ′ , {\displaystyle (fg)'=f'g+fg',} which is the usual product rule and is known ...