Items tagged with derivative derivative Tagged Items Feed

Hi, 


     I've been playing around with the Physics package, and I'm confused on evaluaing derivatives of explicit funcitons of the coordinates. This code below doesnt behave as I would think. I'm trying to define z as a function of X[mu]*X[mu], and take diff(z, X[mu]). You can see that each method d_, diff,  disagree and none are satisfactory ansers. (Maple 2015, Windows 8.1 64-bit, Intel i5 Haswell) 

# Declare coordinates for 2 dimensions, flat space

restart:
with(Physics):
Setup(mathematicalnotation = true, dimension = 2):
Coordinates(X):

# Method 1: Using Define and various differential operators
Define(z):
z :=sqrt(R^2-X[mu]*X[mu]);
d_[mu](z(X));
d_[1](z(X));
diff(z, x1);  #This one is correct
diff(z, X[mu]); # off by 2

# Method #2: Using functions
# Off by a factor of 2
z2 := mu -> sqrt(R^2-X[mu]*X[mu]);
diff(z2(mu), X[mu]); # off by 2

 PhysicsDiffBug.mw

In the following, the diff operator calcuates the derivative correctly, but the D operator doesn't.  A bug?

restart;

f := x -> a[1][2]*x;    # the double index on a[][] is intended

proc (x) options operator, arrow; a[1][2]*x end proc

 

diff(f(x), x);

a[1][2]

 

D(f)(x);

(D(f))(x)

 


Here is a worksheet containing the commands above in case you want to try it yourself: mw.mw

Hello everyone!

I'm pretty new with Maple. I think I've understood the way Maple handles differentiation fairly well, but upon a specific request from my PhD tutor I have to perform a task which is giving me a hard time. 

My question is: is in any way possible to express a derivative of a function or expression in terms of the function itself?
I'll try to explain myself with an example: let f(x) and f'(x) be the function and its first derivative:

Instead of expressing f'(x) in the way shown, I'd like to express it as a function of f(x), such as in the following:

I would apply the same process to the higher order derivatives, if possible.

A huge thank you to whoever will help me!

When I take the derivative of abs(x), I use the chain rule and get this

When I ask Maple to differentiate abs(x), I get this:

I read the help file on "signum", and I expected this to work, but it does not.

 

 

How can I represent signum in normal calculus syntax when working the derivative of functions involving abs(x)?

 

 

 

Determine using determinants the range of values of a (if any) such that
f(x,y,z)=4x^2+y^2+2z^2+2axy-4xz+2yz
has a minimum at (0,0,0).

From the theory, I understand that if the matrix corresponding to the coefficients of the function is positive definite, the function has a local min at the point. But, how do I get the range of values of a such that f is a min? Is this equivalent to finding a such that det(A) > 0?

 

2.

Now modify the function to also involve a parameter b: g(x,y,z)=bx^2+2axy+by^2+4xz-2a^2yz+2bz^2. We determine conditions on a and b such that g has a minimum at (0,0,0).
By plotting each determinant (using implicitplot perhaps, we can identify the region in the (a,b) plane where g has a local minimum.

Which region corresponds to a local minimum?

Now determine region(s) in the (a,b) plane where g has a local maximum.

I don't understand this part at all..

I have the function:   f(x,y) = 1/(sqrt(2*Pi)) * e-1/2(x^2+y^2)

I need to take the total derivative of this function in maple, but I don't know the syntax.

I've plotted the graph for this max function. Is there any way I can find the points of discontinuity in general and then use that to compute the derivatives at points where it exists?

Hi Everyone,

I have an expression that contains a second order derivative: EXPR:=ay''(x) + bz'(x) + cf'(x)+... The variable y  obeys an ordinary differential equation, y''(x) = f(y,x). I would like to replace the second order deriavtive in my expression with f(y,x). So far I have tried applyrule([y''(x)=f(y,x)],EXPR), subs(y''(x)=f(y,x),EXPR) and algsubs(y''(x)=f(y,x),EXPR) and nothing seems to work. Any helpful suggestions?

 

Hello,

 

could you help me solve this error ? I don't understand what it means.

 


> eq3:=diff(x(t),t,t)+Gamma*diff(x(t),t)+omega[0]^2*(x(t)-(diff(x(t),t,t)+Gamma*diff(x(t),t)+omega[0]^2*x(t)+omega[0]^2*X[0])/omega[0]^2) = -omega[0]^2*X[0]:
> dsolve(eq3);
Warning, it is required that the numerator of the given ODE depends on the highest derivative. Returning NULL.

 

Thanks.

Hello i want to sort according to u derivatives (k) system.  And finding determining equations system and solving this system. Thank you very much.  

restart

with(PDEtools)

[CanonicalCoordinates, ChangeSymmetry, CharacteristicQ, CharacteristicQInvariants, ConservedCurrentTest, ConservedCurrents, ConsistencyTest, D_Dx, DeterminingPDE, Eta_k, Euler, FromJet, InfinitesimalGenerator, Infinitesimals, IntegratingFactorTest, IntegratingFactors, InvariantEquation, InvariantSolutions, InvariantTransformation, Invariants, Laplace, Library, PDEplot, PolynomialSolutions, ReducedForm, SimilaritySolutions, SimilarityTransformation, Solve, SymmetrySolutions, SymmetryTest, SymmetryTransformation, TWSolutions, ToJet, build, casesplit, charstrip, dchange, dcoeffs, declare, diff_table, difforder, dpolyform, dsubs, mapde, separability, splitstrip, splitsys, undeclare]

(1)

U := diff_table(u(x, y, t))

table( [(  ) = u(x, y, t) ] )

(2)

declare(U[])

u(x, y, t)*`will now be displayed as`*u

(3)

pde := diff(U[t]-(3/2)*U[x]-6*U[]^2*U[x]+U[x, x, x], x)+U[y, y] = 0

diff(diff(u(x, y, t), t), x)-(3/2)*(diff(diff(u(x, y, t), x), x))-12*u(x, y, t)*(diff(u(x, y, t), x))^2-6*u(x, y, t)^2*(diff(diff(u(x, y, t), x), x))+diff(diff(diff(diff(u(x, y, t), x), x), x), x)+diff(diff(u(x, y, t), y), y) = 0

(4)

NULL

w := phi(x, y, t, U[])

phi(x, y, t, u(x, y, t))

(5)

w*(-12*U[x]^2-12*U[]*U[x, x])+12*w*U[x]^2+12*U[]*w*U[x, x]+(diff(w, x, x))*(-3/2-6*U[]^2)+diff(diff(w, t), x)+diff(w, y, y)+diff(w, x, x, x, x)-lambda*(diff(U[t]-(3/2)*U[x]-6*U[]^2*U[x]+U[x, x, x], x)+U[y, y])

-lambda*(diff(diff(u(x, y, t), t), x)-(3/2)*(diff(diff(u(x, y, t), x), x))-12*u(x, y, t)*(diff(u(x, y, t), x))^2-6*u(x, y, t)^2*(diff(diff(u(x, y, t), x), x))+diff(diff(diff(diff(u(x, y, t), x), x), x), x)+diff(diff(u(x, y, t), y), y))+(D[1, 1, 1, 1](phi))(x, y, t, u(x, y, t))+(D[1, 1, 1, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x))+((D[1, 1, 1, 4](phi))(x, y, t, u(x, y, t))+(D[1, 1, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x)))*(diff(u(x, y, t), x))+(D[1, 1, 4](phi))(x, y, t, u(x, y, t))*(diff(diff(u(x, y, t), x), x))+((D[1, 1, 1, 4](phi))(x, y, t, u(x, y, t))+(D[1, 1, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x))+((D[1, 1, 4, 4](phi))(x, y, t, u(x, y, t))+(D[1, 4, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x)))*(diff(u(x, y, t), x))+(D[1, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(diff(u(x, y, t), x), x)))*(diff(u(x, y, t), x))+2*((D[1, 1, 4](phi))(x, y, t, u(x, y, t))+(D[1, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x)))*(diff(diff(u(x, y, t), x), x))+(D[1, 4](phi))(x, y, t, u(x, y, t))*(diff(diff(diff(u(x, y, t), x), x), x))+((D[1, 1, 1, 4](phi))(x, y, t, u(x, y, t))+(D[1, 1, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x))+((D[1, 1, 4, 4](phi))(x, y, t, u(x, y, t))+(D[1, 4, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x)))*(diff(u(x, y, t), x))+(D[1, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(diff(u(x, y, t), x), x))+((D[1, 1, 4, 4](phi))(x, y, t, u(x, y, t))+(D[1, 4, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x))+((D[1, 4, 4, 4](phi))(x, y, t, u(x, y, t))+(D[4, 4, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x)))*(diff(u(x, y, t), x))+(D[4, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(diff(u(x, y, t), x), x)))*(diff(u(x, y, t), x))+2*((D[1, 4, 4](phi))(x, y, t, u(x, y, t))+(D[4, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x)))*(diff(diff(u(x, y, t), x), x))+(D[4, 4](phi))(x, y, t, u(x, y, t))*(diff(diff(diff(u(x, y, t), x), x), x)))*(diff(u(x, y, t), x))+3*((D[1, 1, 4](phi))(x, y, t, u(x, y, t))+(D[1, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x))+((D[1, 4, 4](phi))(x, y, t, u(x, y, t))+(D[4, 4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x)))*(diff(u(x, y, t), x))+(D[4, 4](phi))(x, y, t, u(x, y, t))*(diff(diff(u(x, y, t), x), x)))*(diff(diff(u(x, y, t), x), x))+3*((D[1, 4](phi))(x, y, t, u(x, y, t))+(D[4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x)))*(diff(diff(diff(u(x, y, t), x), x), x))+(D[4](phi))(x, y, t, u(x, y, t))*(diff(diff(diff(diff(u(x, y, t), x), x), x), x))+(D[2, 2](phi))(x, y, t, u(x, y, t))+(D[2, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), y))+((D[2, 4](phi))(x, y, t, u(x, y, t))+(D[4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), y)))*(diff(u(x, y, t), y))+(D[4](phi))(x, y, t, u(x, y, t))*(diff(diff(u(x, y, t), y), y))+(D[1, 3](phi))(x, y, t, u(x, y, t))+(D[3, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x))+((D[1, 4](phi))(x, y, t, u(x, y, t))+(D[4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x)))*(diff(u(x, y, t), t))+(D[4](phi))(x, y, t, u(x, y, t))*(diff(diff(u(x, y, t), t), x))+((D[1, 1](phi))(x, y, t, u(x, y, t))+(D[1, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x))+((D[1, 4](phi))(x, y, t, u(x, y, t))+(D[4, 4](phi))(x, y, t, u(x, y, t))*(diff(u(x, y, t), x)))*(diff(u(x, y, t), x))+(D[4](phi))(x, y, t, u(x, y, t))*(diff(diff(u(x, y, t), x), x)))*(-3/2-6*u(x, y, t)^2)+12*u(x, y, t)*phi(x, y, t, u(x, y, t))*(diff(diff(u(x, y, t), x), x))+12*phi(x, y, t, u(x, y, t))*(diff(u(x, y, t), x))^2+phi(x, y, t, u(x, y, t))*(-12*(diff(u(x, y, t), x))^2-12*u(x, y, t)*(diff(diff(u(x, y, t), x), x)))

(6)

k := simplify(%)

-(3/2)*(D[1, 1](phi))(x, y, t, u(x, y, t))+(D[1, 3](phi))(x, y, t, u(x, y, t))+(D[2, 2](phi))(x, y, t, u(x, y, t))+(D[1, 1, 1, 1](phi))(x, y, t, u(x, y, t))+4*(D[1, 1, 1, 4](phi))(x, y, t, u(x, y, t))*(D[1](u))(x, y, t)+6*(D[1, 1, 4](phi))(x, y, t, u(x, y, t))*(D[1, 1](u))(x, y, t)+4*(D[1, 4](phi))(x, y, t, u(x, y, t))*(D[1, 1, 1](u))(x, y, t)+(D[4](phi))(x, y, t, u(x, y, t))*(D[1, 1, 1, 1](u))(x, y, t)+2*(D[2, 4](phi))(x, y, t, u(x, y, t))*(D[2](u))(x, y, t)+(D[4](phi))(x, y, t, u(x, y, t))*(D[2, 2](u))(x, y, t)+(D[3, 4](phi))(x, y, t, u(x, y, t))*(D[1](u))(x, y, t)+(D[4](phi))(x, y, t, u(x, y, t))*(D[1, 3](u))(x, y, t)-3*(D[1, 4](phi))(x, y, t, u(x, y, t))*(D[1](u))(x, y, t)-(3/2)*(D[4](phi))(x, y, t, u(x, y, t))*(D[1, 1](u))(x, y, t)-lambda*(D[1, 3](u))(x, y, t)+(3/2)*lambda*(D[1, 1](u))(x, y, t)-lambda*(D[1, 1, 1, 1](u))(x, y, t)-lambda*(D[2, 2](u))(x, y, t)+6*(D[1, 1, 4, 4](phi))(x, y, t, u(x, y, t))*(D[1](u))(x, y, t)^2+4*(D[1, 4, 4, 4](phi))(x, y, t, u(x, y, t))*(D[1](u))(x, y, t)^3+(D[4, 4, 4, 4](phi))(x, y, t, u(x, y, t))*(D[1](u))(x, y, t)^4+3*(D[4, 4](phi))(x, y, t, u(x, y, t))*(D[1, 1](u))(x, y, t)^2+(D[4, 4](phi))(x, y, t, u(x, y, t))*(D[2](u))(x, y, t)^2+(D[3](u))(x, y, t)*(D[1, 4](phi))(x, y, t, u(x, y, t))-(3/2)*(D[4, 4](phi))(x, y, t, u(x, y, t))*(D[1](u))(x, y, t)^2-6*(D[1, 1](phi))(x, y, t, u(x, y, t))*u(x, y, t)^2+12*lambda*u(x, y, t)*(D[1](u))(x, y, t)^2+6*lambda*u(x, y, t)^2*(D[1, 1](u))(x, y, t)+12*(D[1](u))(x, y, t)*(D[1, 4, 4](phi))(x, y, t, u(x, y, t))*(D[1, 1](u))(x, y, t)+6*(D[1](u))(x, y, t)^2*(D[4, 4, 4](phi))(x, y, t, u(x, y, t))*(D[1, 1](u))(x, y, t)+4*(D[1](u))(x, y, t)*(D[4, 4](phi))(x, y, t, u(x, y, t))*(D[1, 1, 1](u))(x, y, t)+(D[3](u))(x, y, t)*(D[4, 4](phi))(x, y, t, u(x, y, t))*(D[1](u))(x, y, t)-12*(D[1, 4](phi))(x, y, t, u(x, y, t))*(D[1](u))(x, y, t)*u(x, y, t)^2-6*(D[4, 4](phi))(x, y, t, u(x, y, t))*(D[1](u))(x, y, t)^2*u(x, y, t)^2-6*(D[4](phi))(x, y, t, u(x, y, t))*(D[1, 1](u))(x, y, t)*u(x, y, t)^2

(7)

frontend(coeff, [k, U[x]^2]);

0

(8)

frontend(coeff, [k, U[x]*U[x, x]])

Error, invalid input: coeff received O*O, which is not valid for its 2nd argument, x

 

NULL


Download det.eq..mw

 

with(plots):

a:=polarplot(3-3*cos(theta),theta=0..2*Pi):

c:=plot((3*sqrt(2)+3)/2 + ((-3*sqrt(2))/(-3*sqrt(2)-6))*(x+((3*sqrt(2)+3)/2)),x=-10..10):

display(a,c,view=[-10..10,-10..10]);

 

a:= is the polar plot of the cardiod (3-3cos(theta)


In order to plot the tangent line to the cardiod in theta= 3Pi/4, I find the point (x,y) in rectangular coord x=(3-3cos(theta)cos(theta) and y=(3-3cos(theta)sin(theta); then I find the derivative of dx/dy=

[(3-3*cos(theta)*cos(theta)+sin(theta)(3sin(theta)]/[-(3-3cos(theta)sin(theta)+cos(theta)(3sin(theta)]and from here I get the slope.So I can plot c:= tangent line to the cardiod in 3Pi/4.

How can I avoid having to convert everyting to rectangular coords, and plot the tangent line in polars?

 

 

Hi All,

I have a problem with regard to partial differential equations. I am using Lagrangian dynamics for a problem. First i have a function First i defined a function with two speeds of angles (first derivatives):

ODE := 5*(diff(theta1(t), t))+diff(theta2(t), t). This gives:

Now this gives an output. Lagrange (just a simple example now) demands that i now derive the obtained function with regard to the first derivative of theta1. In this case, the answer i want is 5. Now, if i give the command: 

diff(ODE, diff(theta1(t),t)), maple says go home. Does anybody know how to solve this? I have been searching for a solution all afternoon.

 

Thnx in advance!

Hi everyone,

I have a question regarding the derivation of tensors/matrices.
Let's assume for simplicity, that I have a vector (6x1) s and a matrix A (6x6)defining Transpose(s)*Inverse(A)*s. From this function I want to calculate the derivative w.r.t. s. My approach would be

restartwith(Physics):
with(LinearAlgebra):
Define(s,A)

Diff(
Transpose(s)*Inverse(A)*s, s)

As a result I get

though I'd rather expect something like Inverse(A)*s + Transpose(s)*Inverse(A)

Now as I'm pretty new to Maple, I can imagine that my approach is wrong, but I don't know any better and can't seem to get any information out of the help documents.

Thanks in advance for any of your suggestions!

Hi

I need you to help me in writting procedure with input "r" ( order of derivative) and some coefficients seq(alpha[i],i=1..N).  My code work very well, need only put all the element in procedure with output The Taylor series obtained in last line of my code and the order of error.  I want the procedure return the coefficients beta used in the series and the order of Error  and the coefficients beta[i]

May thinks.

Second_Question.mw

 

I will not modify my previous question: only I add this remark. I tried to write these lines of procedure.
Add_display.mw

I need only to add( if the procedure is true) these lines in my procedure.  These lines gives the degree of error computed in the code, but when I put these lines in the codes, there is an error. Thanks for your help.

Degree := degree(Error,stepsize);
    if (showorder) then:
       print(cat(`This stencil is of order `,Degree));
    fi:
    if (showerror) then:
       print(cat(`This leading order term in the error is `,Error));
    fi:
    convert(D[r$n](f)(vars) = stencil,diff);

 

 

 

Dear All, I need your help to plot the numerical solution. many thanks.

The variable t in [0,T], x in [0,1], b in [0,2].

Difference finie for waves equation is :

pde:=diff(u(x, y,t), t$2) = c^2*(diff(u(x, y,t),x$2)+diff(u(x,y,t),y$2));

i: according to x, j according to y, and k according to t.

u[i,j,k+1]=2*u[i,j,k]-u[i,j,k-1]+(c*dt/dx)^2*(u[i-1,j,k]-2*u[i,j,k]+u[i+1,j,k])+ (c*dt/dy)^2*(u[i,j-1,k]-2*u[i,j,k]+u[i,j+1,k])

 

Boundary condition: u(t=0)=1, diff(u(x,y,t),t=0)=0, and the normal derivative on the boundary of Omega =0.

How can solve this problem and plot the numerical solution.

 

 

 

1 2 3 4 5 6 7 Last Page 1 of 22