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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 symmetric derivative at a given point equals the arithmetic mean of the left and right derivatives at that point, if the latter two both exist. [1] [2]: 6 Neither Rolle's theorem nor the mean-value theorem hold for the symmetric derivative; some similar but weaker statements have been proved.
The last formula, where summation starts at i = 3, follows easily from the properties of the exterior product. Namely, dx i ∧ dx i = 0. Example 2. Let σ = u dx + v dy be a 1-form defined over ℝ 2. By applying the above formula to each term (consider x 1 = x and x 2 = y) we have the sum
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011.
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.
Since Mathieu's equation is a second order differential equation, one can construct two linearly independent solutions. Floquet's theory says that if a {\displaystyle a} is equal to a characteristic number, one of these solutions can be taken to be periodic, and the other nonperiodic.