CS1231 - Lecture 09
... We have proven that f is a bijection. Let’s pause for a moment to consider this propositoin that we have just proven. Z is infinite. Z+ is infinite. Which is more infinite? Most would answer intuitively that Z contains more elements than Z+. But your intuition must be in subjection to logic. And log ...
... We have proven that f is a bijection. Let’s pause for a moment to consider this propositoin that we have just proven. Z is infinite. Z+ is infinite. Which is more infinite? Most would answer intuitively that Z contains more elements than Z+. But your intuition must be in subjection to logic. And log ...
Stochastic Differential Equations (SDEs)
... and it constitutes a survey paper on the title subject. Stochastic differential equations (SDEs) appear today as a modeling tool in several sciences as telecommunications, economics, finance, biology and quantum field theory. A stochastic differential equation is essentially a classical differential ...
... and it constitutes a survey paper on the title subject. Stochastic differential equations (SDEs) appear today as a modeling tool in several sciences as telecommunications, economics, finance, biology and quantum field theory. A stochastic differential equation is essentially a classical differential ...
Exact Differential Equations
... Proof We first prove that exactness implies the validity of Equation (1.9.5). If the differential equation is exact, then by definition there exists a potential function φ(x, y) such that φx = M and φy = N. Thus, taking partial derivatives, φxy = My and φyx = Nx . Since My and Nx are continuous in R, ...
... Proof We first prove that exactness implies the validity of Equation (1.9.5). If the differential equation is exact, then by definition there exists a potential function φ(x, y) such that φx = M and φy = N. Thus, taking partial derivatives, φxy = My and φyx = Nx . Since My and Nx are continuous in R, ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.