Line integral, jacobian...

[Han]

(02-02-2015 08:56 PM)salvomic Wrote:  hi all,
I need some hints to use with Prime:
curve integral (for scalar function by a curve)
line integral (vectorial camp by a curve)
flux integral

and also something to treat with Jacobian Matrix (where line are gradients).

Any programs, functions? hints to create my own program easily?

Thank a lot in advance!

Salvo

You should be able to do all of these in the CAS. For example, if C is the path from (0,0) to (1,1) then C is described by \( \vec{r}(t) = t\vec{i} + t\vec{j} \) where \( 0 \le t \le 1 \). If the force F is defined by \( \vec{F} = (3x-2y)\vec{i} + (4x+1)\vec{j} \) then
\[ \int \vec{F}\cdot d\vec{r} \]
can be computed by:

r:=[t,t];
dr:=diff(r,t);
f:=[3*x-2*y, 4*x+1];
ft:=subst(f,[x,y]=r);
int(dot(ft,dr),t,0,1);

If \( \vec{F} \) is path-independent (i.e. a gradient field; also known as conservative fields) then one can simply apply the fundamental theorem of calculus for vectors. For example, if
\[ \mathrm{grad}(f) = 2xy \vec{i} + (x^2+8y^3)\vec{j} \]
then you can do something like:

f1:=2*x*y; f2:=x^2+8y^3;

check if we actually have a real gradient field:

diff(f1,y) - diff(f2,x);

(simplify if needed; gradient field means difference should be 0). If we have a gradient field, i.e. \( \vec{F} = \mathrm{grad}(f) \), then we can find f by:

f:=int(f1,x);
dg:=diff(f,y)-f2;
g:=int(dg,y);
f:=simplify(f+g); // can optionally leave out simplify()
p:=[0,0];
q:=[1,1];
subst(f,[x,y]:=q)-subst(f,[x,y]=p);

Edit: A CAS program is really nothing more than copying the manual commands you use in the CAS view into a CAS-program, and making some minor edits and slapping #cas and #end around everything. So I recommend playing around with the CAS view, and once you have what you want, copy it into a program.