In this paper we point out that the proofs of Chain Rule in many intensively used textbooks are not strict an construct anexa mple of a composite function f(u) which is differentiable with respects t o the independe...In this paper we point out that the proofs of Chain Rule in many intensively used textbooks are not strict an construct anexa mple of a composite function f(u) which is differentiable with respects t o the independednt variable u, but is not differentiable with respect to the dependent variable u=g(x). A strict proof of Chain Rule is presented. Incon sistency of the form and content of Chain Rule is disclosed.展开更多
Fractional calculus is a powerful tool for modeling nonlinear systems.It is necessary to discuss the basic properties of fractional order before solving a fractional order model.Using the formula of power function def...Fractional calculus is a powerful tool for modeling nonlinear systems.It is necessary to discuss the basic properties of fractional order before solving a fractional order model.Using the formula of power function defined by local fractional derivative and the chain rule to calculate a compound function,the results are inconsistent.This shows that the chain rule of local fractional derivatives similar to classical calculus is suspicious,and fractional complex transformation based on the chain rule is also suspicious and needs further discussion.In order to overcome this inconsistency,an improved definition of local fractional derivative,which can be regarded as a fractal derivative,is proposed based on the results derived from the relationship between the mass function and the Hausdorff measure.展开更多
It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition...It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)A(s)ds,A(t)]=0.A natural question is whether this is true without the commutativity condition.To give a definite answer to this question,we present two classes of illustrative examples of coefficient matrices,which satisfy the chain rule d/dt exp(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.展开更多
文摘In this paper we point out that the proofs of Chain Rule in many intensively used textbooks are not strict an construct anexa mple of a composite function f(u) which is differentiable with respects t o the independednt variable u, but is not differentiable with respect to the dependent variable u=g(x). A strict proof of Chain Rule is presented. Incon sistency of the form and content of Chain Rule is disclosed.
基金Major Science and Technology Project in Shanxi Province of China(Nos.20181101008 and 20181102015)Supplementary Platform Project of“1331”Project in Shanxi Province in 2018,China。
文摘Fractional calculus is a powerful tool for modeling nonlinear systems.It is necessary to discuss the basic properties of fractional order before solving a fractional order model.Using the formula of power function defined by local fractional derivative and the chain rule to calculate a compound function,the results are inconsistent.This shows that the chain rule of local fractional derivatives similar to classical calculus is suspicious,and fractional complex transformation based on the chain rule is also suspicious and needs further discussion.In order to overcome this inconsistency,an improved definition of local fractional derivative,which can be regarded as a fractal derivative,is proposed based on the results derived from the relationship between the mass function and the Hausdorff measure.
基金supported in part by the Established Researcher Grant and the CAS Faculty Development Grant of the University of South Florida,Chunhui Plan of the Ministry of Education of China,Wang Kuancheng Foundation,the National Natural Science Foundation of China(Grant Nos.10332030,10472091 and 10502042)the Doctorate Foundation of Northwestern Polytechnical University(Grant No.CX200616).
文摘It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)A(s)ds,A(t)]=0.A natural question is whether this is true without the commutativity condition.To give a definite answer to this question,we present two classes of illustrative examples of coefficient matrices,which satisfy the chain rule d/dt exp(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.
