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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 ...
Complete this to a basis {r u,r v,n}, by selecting a unit vector n normal to the surface. It is possible to express the second partial derivatives of r (vectors of R 3 {\displaystyle \mathbb {R^{3}} } ) with the Christoffel symbols and the elements of the second fundamental form.
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 second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
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 covariant derivative of a function (scalar) ... defined as the -trace of the second fundamental form. Then ~ = (()). Note that this transformation ...
Sometimes other equivalent versions of the test are used. In cases 1 and 2, the requirement that f xx f yy − f xy 2 is positive at (x, y) implies that f xx and f yy have the same sign there. Therefore, the second condition, that f xx be greater (or less) than zero, could equivalently be that f yy or tr(H) = f xx + f yy be greater (or less ...
A cubic of the form , = {(,): =}, where , are complex numbers with , cannot be rationally parameterized. [1] Yet one still wants to find a way to parameterize it. For the quadric = {(,): + =}; the unit circle, there exists a (non-rational) parameterization using the sine function and its derivative the cosine function: : /, (, ).