Hello people in mapleprimes,

I have an equation called aa in the following.

alias(`δx`=dx, `Δx`=Dx,`Δy`=Dy,`Δz`=Dz):
aa:=Dz=f(x+Dx,y+Dy)-f(x,y);

As for modification of this expression, I ask your favor to teach me.

Then, I want to change this aa to D[1](f)(x+theta__1*Dx,y+Dy)*Dx+D[2](f)(x,y+theta__2*Dy)*Dy.

But, to do so, I have to split aa into the one including -f(x,y+Dy)+f(x,y+Dy) between two terms

of aa.

But, as maple cancels these terms, I can't do so. How can I insert two terms, then

obtain the expression  D[1](f)(x+theta__1*Dx,y+Dy)*Dx+D[2](f)(x,y+theta__2*Dy)*Dy?

Thanks in advance.

taro

Hello people in mapleprimes,

I have a question. I hope someone give an answer to me.

rho:=sqrt((x-a)^2+(y-b)^2);

limit(x-a+y-2,rho=0);

brings an error

Error, invalid input: limit expects its 2nd argument, p, to be of type Or(name = algebraic, set(name = algebraic)), but received ((x-a)^2+(y-b)^2)^(1/2) = 0

Isn't other way than the following?

limit(x-a+y-2,{x=a,y=b});

Best wishes.

taro

#Hello people in Mapleprimes,

#After

restart;interface(typesetting=extended);

diff(f(x),x);

#shows f'(x).

#But,

diff(f(t1),t1);

#shows just df(t1)/dt1, not f'(t).

#Can't I show f'(t1) not df(t1)/dt1?

#Is this a way peculiar to Maple?

#I hope someone could give me some hints.

#Best wishes.

#taro

Hello people in mapleprime,

Though I wrote the title as Fundamental theorem of calculus,

what I am considering is just how to continue the chain of codes in calculation.

restart;

#I defined F__0 as

F__0:=x->Int(f(t),t=a..x);

#Then, the difference between a primitive function of f(x), F(x), and F__0 is no more than a constant C, so I write.

bb:=F(x)-F__0(x)=C;

#Then, substituting "a" into equation "bb", I obtain the value of F(a)

bb1:=subs(x=a,bb):cc:=simplify(%);

#Then, I substituted the value of C in "bb1" into "bb,"  obtaining the following "cc1."

cc1:=subs(isolate(cc,C),bb);

#And, then, I isolated the term of Int(f(t),t=a..x)  in cc1,

dd:=isolate(cc1,Int(f(t), t = a .. x));

#And, then, I substitute x=b into the outcome of dd, and obtain the final equation.

subs(x=b,dd);

Surely, with the above code, I could get the fundamental theorem. But, it looks in a little roundabout way.

So, I thought I would ask here about whether there aren't any better ways to do the fundamentally the same thing or

hints to improve the above code.

Please teach me about this.

Thanks in advance.

taro

Hello people in the mapleprimes,

I have a question, so I hope someone give me answers to it.

I calculated for the solution of the follwing differential equation.

restart
b:=diff(y(x),x)+a*y(x)=f(x);#where a and f(x) is not specified.
dsolve(b,y(x));

subs({f(x)=exp(x),a=2},%);where f(x) and a are specified.

c:=value(%);

The solution of the above was

y(x) = (1/3)*exp(x)+_C1/(exp(x))^2,  (A)

where please note that the second term takes

the form of fraction _C/(exp(x))^2.

On the other hand, next I calculated the following differential equation where f(x) and a are specified from the start.

restart
d:=diff(y(x),x)+2*y(x)=exp(x);

dsolve(b,y(x));

Then,

y(x) = (1/3)*exp(x)+exp(-2*x)*_C1  (B)

was the obtained solution.

Each (A) and (B) are the same substantially mathematically. But, for Maple, the variable powered to minus brabra

is not the same as one over variable powered to brabra, so that (A) and (B) takes different forms, and maple will see them 

different with each other.

  Surely, with algsubs, algsubs(_C1/(exp(x))^2=exp(-2*x)*_C1,c) transforms (A) to (B).

But, I want to know whether there are some other ways than that  to modify (A) to (B).

If there are any good ways for it, I will be happy if you teach them to me.

Thanks in advance.

taro