Questions tagged [differentials]

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Product of different differential operators

I need to define a multivariable differential equation of the type $$ \mathcal{D}=\Theta_1^2-z_1(-\Theta_1+\Theta_2+\Theta_3+\Theta_4)(1+\Theta_1+\Theta_2+\Theta_3+\Theta_4)\\ \;\;\;\;\;\;\;\;\;\;\;\;\...
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  • 21
1 vote
1 answer
61 views

How to perform the first and second derivatives of Nth polynomials?

Suppose  $P(x,y)=\displaystyle\sum^{N}_{i+j=0}\alpha_{ij}x^{i}y^{i} \equiv \alpha_{00}+\alpha_{10}x+\alpha_{01}y+a_{11}xy+\ldots+a_{0N}y^{N}$ is a multivariable polynomial of $N$th degree. I want to ...
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5 votes
3 answers
129 views

What's the Mathematica command for $\frac{\partial^a}{\partial n^a}\binom{n}{k}$?

What's the Mathematica command for $$f(a,n,k)=\frac{\partial^a}{\partial n^a}\binom{n}{k}$$ I tried f[a_,n_,k_]:=D[Binomial[n,k], {n,a}] but didn't work. To make ...
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0 votes
1 answer
47 views

How to evaluate these derivatives

I'm trying to evaluate the following functions S1,S2 and their derivatives: As you can see the S1 and it's derivatives it's working but for the S2 and it's derivatives I'm getting the error displayed ...
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  • 103
0 votes
1 answer
109 views

Second order differential operator in Mathematica

I am trying to write the code for the following expression in Mathematica: $$\partial_{x} e^{-g x}=-g e^{-g x}+e^{-g x} \partial_{x}\,.$$ I was able to do so with the help of the following links with ...
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  • 3
0 votes
1 answer
74 views

Mathematica: Partial derivative with respect to tensors

Anyone know how to do the partial derivative of a tensor in d dimensions on Mathematica ? I want to implement something which will calculate directly like that :
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1 vote
1 answer
67 views

How can I make sure if the given curve is monotonic with respect to $x$ and $y$?

I have two functions $ f(x,y) $ and $ g(x,y) $ over the domains $ 1<x< 2$ and $0<y<\pi$. I use ContourPlot to ses those values of $(x,y)$ for which $g(...
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  • 463
-1 votes
1 answer
99 views

Van der Pol's equation [closed]

Consider the van der Pol equation below: (x′′)+c(x^2−1)(x′)+(x)=0 , which governs the flow of current in a vacuum tube with three internal elements. It can be written as the system x′= y , y′= -x-c(x^...
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0 votes
0 answers
58 views

Exclude a term in a differential equation

How can I exclude this term (The red box) from derivation to make it consider it as a constant (Just a constant to be multiplied ), I am getting the differentiation for the whole equation but I want a ...
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0 answers
66 views

Derivative with respect to function of any argument

I have the expression L = v1[k1] * M[k1, k2] * v2[k2]. v1, M, and ...
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0 answers
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Table with two indices does not work properly

I am a beginner in computational mathematics so apologies in advance for my probably poor code writing skills. I am using the package << matrixEDC.m to do some exterior algebra computations. ...
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0 votes
1 answer
213 views

Partial derivatives of this function

I have a function $y(x)$ which is the solution of the initial value problem - $y = \frac{1-y}{A} - \frac{y.|r(x)+m(x)|^2}{B}$ with an initial condition $y_0=1$. I solved for $y(x)$ using the following ...
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  • 109
1 vote
2 answers
133 views

How to make a change of variable $y=x^2$ in the differential operator $\frac{d^2}{d y^2}$?

I have this differential operator $$\frac{d^2}{d y^2}\qquad (1)$$ I want to make a change of variable $y=x^2$ in $(1)$ so that I get this differential operator $$ \frac{1}{4 x^2} \frac{d^2 }{d x^...
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  • 335
1 vote
1 answer
57 views

solving differential equation with matrices

I have to solve two coupled differential equations: ...
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0 votes
1 answer
94 views

Transformation of Differential Operators

I would like to gain some knowledge about how to transform differential operators to different coordinate systems using Mathematica. I know that the Laplacian ...
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  • 831
2 votes
2 answers
168 views

How to solve these two differential equations

I have two differential equations and I try to use function DSolve to solve them together. ...
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  • 121
0 votes
1 answer
143 views

How can i solve and plot this non linear ODEs usin NDSOLVE?

I am trying to solve the following non-linear ODEs ...
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3 votes
2 answers
220 views

How calculate integral of differential form [duplicate]

I've this differential form: w = (e^y + z*e^x)dx + ((x + 1) e^y + e^z)dy + ((y + 2) e^z + e^x)dz It is sure exact, so I'm trying to find a primitive of w. ...
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  • 121
0 votes
1 answer
58 views

Problem in table [duplicate]

NDSolve[{t,Evaluate[{ode}\.Sol]},{t,0,10,1}]//TableForm After executing this with appropriate equations I get solution in the tabular form but when I try to ...
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2 votes
1 answer
161 views

Partial Differentiation in Mathematica, syntax error? [closed]

I am trying to differentiate some random function using Mathematica at particular nodes, I am though not being able to implement this without being retrieved the following error message: D: ...
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  • 87
2 votes
1 answer
37 views

How to differentiate a composite function which includes unknown functions?

Let $f$ be a $C^2$ function from $\mathbb{R}^3$ to $\mathbb{R}$. Let $\Phi$ be the function from $(0, \infty) \times [0, \pi] \times [0, 2 \pi]$ to $\mathbb{R}^3$ such that $\Phi(r, \theta, \phi) = (r ...
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1 vote
1 answer
93 views

How to write f[ t[i],...,t[i+r] ]=f [t[i+1],...,t[i+r] ]- f[ t[i] ,...,t[i+r-1] ] in Mathematica for 0 <= i <= n , 0<= r <= n [closed]

For[[[[[[[[[[[Do[]]]]]]]]]]]]]
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0 votes
0 answers
77 views

Smooth approximation near a non differentiable point

Let $f:\mathbb{R}_+\rightarrow \mathbb{R}$ be a function differentiable for $x>0$ but non differentiable at $x=0$ (for instance $f=\sqrt{\cdot}$) and $g$ be a polynomial function. I know how to ...
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  • 442
0 votes
0 answers
35 views

Differential equations system's solutions differ when one of the constant is 0

I am solving the following system of equations with Mathematica. $u'=-k_{1}ux+k_{2}x^2, x'=k_{1}ux-k_{2}x^2$ and I obtain the solution for x: $x(t)=\frac{e^{k_{1}C_{1}(t+C_{2})}k_{1}C_{1}}{-1+e^{k_{1}...
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2 votes
1 answer
56 views

How to differentiate a function that is a function of other functions?

I am trying to differentiate the function epsilon1 with respect to u, but why this does not give me any output? ...
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0 votes
2 answers
126 views

total differential of a subscripted variable

During use of sub-scripted variables, I came across the following issue Dt[Subscript[x,0]] gives Why is mathematica treating x and not x$_0$ as the ...
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  • 902
4 votes
0 answers
91 views

Computing differentials on a manifold

Consider $\phi:SO(3) \to \mathbb{R}^3$, $R \mapsto (R^\top e_3)\times e_3$ where $R$ is a real $3\times 3$ orthogonal matrix and $e_3 = [0\ 0\ 1]^\top$. Can Mathematica compute the differential of $\...
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  • 13.6k
2 votes
1 answer
41 views

Help with evaluation of the differential of a BSplineFunction?

It can be useful to generate a smooth random function over $[-L,L]$ with BSplineFunction: ...
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  • 1,197
2 votes
2 answers
209 views

Vector calculus with index notation

How would you calculate this in Mathematica? $$\sum_i\partial_i\partial_i G_{jk}$$ $$G_{jk}=\bigg(\frac{\delta_{jk}}{r}+\frac{(x_j-y_j)(x_k-y_k)}{r^3}\bigg)$$ where $r=|\boldsymbol{x}- \boldsymbol{...
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3 votes
2 answers
119 views

Defining a differential operator that acts on a non-commutative basis

Given a non-commutative basis ${x_0,x_1,x_2}$ I'd like to define a differential operator that acts as so $$ \Delta_i (\sum_{n=0}^\infty c_n x_i^n) = \sum_{n=1}^\infty c_n x_i^{n-1}, \quad \Delta_i ...
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  • 203
0 votes
0 answers
183 views

Transform a field with derivatives

Given a function of Cartesian coordinates $(x, y)$ in $g(x, y) = \frac{d}{dx}f(x, y)-y\frac{d}{dy}f(x, y)$, I want to transform this expression to polar coordinates $(r,\phi)$. I saw a solution using ...
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0 votes
0 answers
517 views

Commutator of differential operators

Let $P_x = \frac{\hbar}{i}\frac{d}{dx}$, after specifying the commutator relation symbolically $[X, P_x] = i\hbar$, I can ask Mathematica to calculate commutator algebra. My question: is there a way ...
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0 votes
1 answer
37 views

Function that takes derivative of another function (with any number of arguments) and evaluates it

I would like to supply a function as an argument to another function which then differentiates the function and evaluates it at a set of points. Suppose I want to differentiate this function, ...
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  • 3,077
2 votes
0 answers
191 views

Computing quadratic differential trajectories with Mathematica

There was a question about a particular case of this, Quadratic differentials; seemingly it contained too little information, so let me try again. This will be also a second take on my previous ...
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1 vote
1 answer
100 views

How to define a variable to be a function of all the arguments but one?

I have several variables (coordinates) $x_1, x_2, x_3, x_4, z$ and a lot (21, to be precise) functions $g_i$ which are assumed to depend on $x_1, x_2,x_3, x_4$, but not on $z$. I also have a ...
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1 vote
0 answers
53 views

Symbolic differentiation on a function of matrices and vectors

Let $I_t$, $I$, $V$, $Z$ be vectors and $R$, $B$, $P$, $A$, $\mathcal{K}$, $\mathcal{P}$ be some matrices such that $$ \begin{eqnarray} S&=&-(R I+B^\top V)^\top \mathcal{R}(R I+B^\top V) ...
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  • 445
1 vote
2 answers
144 views

Differentiating nested functions, Differentiate unevaluated function

I have two functions G[x_]:=x^6; S[x_]:=G[x]+x^12 I would like to operate on S[x] with respect to x in three ways, where we take the differentiation as ...
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0 votes
3 answers
112 views

Leave Parts of Formula inactive in partial Differentiation

for thermodynamics, I would like to use a thermodynamic derivative $$ \frac 1 T=\frac{\partial S}{\partial E}\Bigg \vert_V $$i.e. the derivative of the entropy S with respect to the energy E with ...
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1 vote
1 answer
496 views

Does Mathematica understand the concept of infinitesimal increment?

Commonly, in textbooks, differential of differential increment vanishes, good example is how Euler-Lagrange equations are derived using the differential (from step 2 to step 3 here, you can see ...
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  • 113
6 votes
2 answers
245 views

Chain rule when taking derivatives with UpValues

I have a summation function, for which I have defined derivation via Custom`sum /: D[Custom`sum[idx_, a_, b_, exprs_], y_] := Custom`sum[idx, a, b D[expr, y]]; ...
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5 votes
4 answers
219 views

Find $\frac{dy}{dx}$ if $x\sqrt{1+y}+y\sqrt{1+x}=0$

Find $\frac{dy}{dx}$ if $x\sqrt{1+y}+y\sqrt{1+x}=0$ for $-1\leq x\leq 1$ $$\frac{dy}{dx}=\frac{-1}{(1+x)^2}\text{ if }x\neq y$$ If it was like $y=\sqrt{1+x}$ i think i know how to do it ...
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  • 311
1 vote
1 answer
192 views

How to program Mathematica to do the differential operator calculation

I want to create a surface Laplacian under the spheroidal coordinates, Now, I already have the surface gradient defined, but I don't know how to find the Laplacian. Since the surface gradient is too ...
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4 votes
2 answers
1k views

How to define a differential operator in Mathematica? [duplicate]

I want to define an operator $(\partial_{t}+1)^{2}=\partial_{t}\partial_{t}+2\partial_{t}+1$. Then, I want it to act on $t$. My code looks like this: ...
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2 votes
1 answer
123 views

Solving algebraic equation system with partial derivatives

I have a list of equations. Some of them are algebraic equations like 2x == y / (y+1), others contains derivatives in them(e.x. ...
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  • 123
1 vote
0 answers
66 views

Symbolic summation with variable bounds and variable number of indices

I wish to compute compute terms like Sum[f[t[j[1]],t[j[2]],...],{j[1],m},{j[2],n},...] for arbitrary positive integer n and any ...
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  • 21
0 votes
0 answers
58 views

Proving $d(dx)=d^2x$

Is it possible to carry out the following differential in Mathematica? $$d(dx) = d^2x$$ I tried to write: Dt[1*Dt[x]] the result is: ...
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1 vote
1 answer
61 views

Differentiate an Integral-Assumption needed

I already looked into the smaller problems, but I just don't get the clean first derivative, because Mathematica wants to work with imaginary numbers. How do I best supress this here? I tried it with <...
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  • 313
2 votes
1 answer
97 views

How does D work with numerical arguments? [duplicate]

If I define f[x_?NumericQ] := (Sow[x]; SawtoothWave[x]) D[f[x], x] /. x -> 1.1 (* 5.29432 *) numeric differentiation fails, despite the function being smooth ...
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  • 14.7k
13 votes
2 answers
310 views

Differentiation and series expansion of dot product - inconsistent results

Bug introduced in 9.0 or earlier and persisting through 11.3 or later Bug resolved in 12.0 As of 12.0, we have an unevaluated result - inconsistent with the differentiation result, but not invalid. ...
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  • 14.7k
0 votes
1 answer
305 views

How to evaluate and simplify series expansions of long partial derivatives involving trigonometry?

I have 4 maps defined on a plane, i.e. on $\mathbb{R}^{2}$. Let's call them $F,G,H,J$. Let $(s,t) \in \mathbb{R}^{2}$. Define $F(s,t) = (f_{1}(s,t), f_{2}(s,t)).$ Likewise, in components, write $$ G(...
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  • 211