In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
The partial derivative of a function
f
(
x
,
y
,
…
)
{\displaystyle f(x,y,\dots )}
with respect to the variable
x
{\displaystyle x}
is variously denoted by
f
x
′
,
f
x
,
∂
x
f
,
D
x
f
,
D
1
f
,
∂
∂
x
f
,
or
∂
f
∂
x
.
{\displaystyle f'_{x},f_{x},\partial _{x}f,\ D_{x}f,D_{1}f,{\frac {\partial }{\partial x}}f,{\text{ or }}{\frac {\partial f}{\partial x}}.}
Sometimes, for
z
=
f
(
x
,
y
,
…
)
,
{\displaystyle z=f(x,y,\ldots ),}
the partial derivative of
z
{\displaystyle z}
with respect to
x
{\displaystyle x}
is denoted as
∂
z
∂
x
.
{\displaystyle {\tfrac {\partial z}{\partial x}}.}
Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:
f
x
(
x
,
y
,
…
)
,
∂
f
∂
x
(
x
,
y
,
…
)
.
{\displaystyle f_{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).}
The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786) (although he later abandoned it, Carl Gustav Jacob Jacobi reintroduced the symbol in 1841).
Hello,
Given a function like ##z= 3x^2 +2y##, the partial derivative of z w.r.t. x is equal to: $$\frac {\partial z}{\partial x} = 6x$$
Let's consider the point ##(3,2)##. If we sat on top of the point ##(3,2)## and looked straight in the positive x-direction, the slope The slope would be...
Some may say that ##\frac{ \partial g }{ \partial t }## is correct because it is a term in a partial differential equation, but since ##g## is a one variable function with ##t## only, I think ##\frac{ dg }{ dt }## is correct according to the original usage of the derivative and partial...
I am studying a 2D material using tight binding. I calculated density of states using this method. Can I also calculate partial density of states using tight binding?
Hi everyone!
It's about the following task.
Partial molar quantities
a) How are partial molar quantities defined in general?
b) If X is an extensive state variable and X̅ is the associated partial variable, what types of variables must X̅ have?
c) Is the chemical potential of component i in a...
I study genotype-environment associations in alpine species. I frequently see altitude as the sole predictor of partial pressure of oxygen in the literature concerning hypoxia adaptations. However, I understand that partial pressure of oxygen is also influenced by temperature, humidity, and...
Hello everyone,
I seem to be majorly lacking in regards to intuition with partial derivatives. I was studying the Euler-Lagrange equations and realized that my normal intuition with derivatives seems to lead me to contradictory or non sensical interpretations when reading partial derivatives...
In the coursebook the question says:
The reaction below was carried out at a pressure of 10×10⁴ Pa and at constant temperature.
N2 + O2 ⇌ 2NO
the partial pressures of Nitrogen and Oxygen are both 4.85×10⁴ pa
Ccalculate the partial pressure of the nitrogen(ll) oxide, NO(g) at equilibrium.
In...
Looking at pde today- your insight is welcome...
##η=-6x-2y##
therefore,
##u(x,y)=f(-6x-2y)##
applying the initial condition ##u(0,y)=\sin y##; we shall have
##\sin y = u(0,y)=f(-2y)##
##f(z)=\sin \left[\dfrac{-z}{2}\right]##
##u(x,y)=\sin \left[\dfrac{6x+2y}{2}\right]##
From Rand Lectures on Light, we have, in the interaction picture, the equation of motion of the reduced density matrix:
$$i \hbar \rho \dot_A (t) = Tr_B[V(t), \rho_{AB}(t)] = \Sigma_b \langle \phi_b | V \rho_{AB} -\rho_{AB} V | \phi_b \rangle = \Sigma_b \phi_b | \langle V \rho_{AB} | \phi_b...
This is part of the notes;
My own way of thought;
Given;
##U_{xy}=0##
then considering ##U_x## as on ode in the ##y## variable; we integrate both sides with respect to ##y## i.e
##\dfrac{du}{dx} \int \dfrac{1}{dy} dy=\int 0 dy##
this is the part i need insight...the original problem...
Since we are adding numbers produced according to a fixed pattern, there must also be
a pattern (or formula) for finding the sum.
Hi, We use this method to find the ##S_n##. I don't understand how the sum will also be in a pattern. Can someone please explain this line in bold?
Part (A): The matrix is a singular matrix because the determinant is 0 with my calculator.
Part (B): Once I perform Gauss Elimination with my pivot being 0.6 I arrive at the last row of matrix entries which are just 0's. So would this be why Gauss Elimination for partial pivoting fails for this...
I'm stuck on this problem, I've tried to follow techniques for similar questions, namely I seem to be struggling with these questions where I have to use an equation inside an equation. I've attached photos of my process so far, but obviously, I'm not getting the right answer because what I'm...
He draws an n-manifold M, a coordinate chart φ : M → Rn, a curve γ : R → M, and a function f : M → R, and wants to specify ##\frac d {d\lambda}## in terms of ##\partial_\mu##.
##\lambda## is the parameter along ##\gamma##, and ##x^\mu## the co-ordinates in ##\text{R}^n##.
His first equality is...
ok I posted this a few years ago but replies said there was multiplication in it so I think its a mater of format
##\dfrac{\partial u^2}{\partial x\partial y}## is equivalent to ##u_{xy}##
textbook
In my book it is written "Ends of dipole possesses partial charges. Partial charges are always less than the unit electronic charge (1.6×10−19 C)".
Suppose in a double bond(two electron is shared by each atom) or triple bond(three electrons are shared by each atom), can the electronegative atom...
I just started to study thermodynamics and very often I see formulas like this:
$$ \left( \frac {\partial V} {\partial T} \right)_P $$
explanation of this formula is something similar to:
partial derivative of ##V## with respect to ##T## while ##P## is constant.
But as far as I remember...
In the 128 pages of 《A First Course in General Relativity - 2nd Edition》:"The covariant derivative differs from the partial derivative with respect to the coordinates only because the basis vectors change."Could someone give me some examples？I don't quite understand it.Tanks！
Hello, I am trying to solve the following problem:
If ##z=f(x,y)##, where ##x=rcos\theta## and ##y=rsin\theta##, find ##\frac {\partial z} {\partial r}## and ##\frac {\partial z} {\partial \theta}## and show that ##\left( \frac {\partial z} {\partial x}\right){^2}+\left( \frac {\partial z}...
If the right-hand side is zero, then it will be a wave equation, which can be easily solved. The right-hand side term looks like a forced-oscillation term. However, I only know how to solve a forced oscillation system in one dimension. I do not know how to tackle it in two dimensions.
I have...
I am going through this page again...just out of curiosity, how did they arrive at the given transforms?, ...i think i get it...very confusing...
in general,
##U_{xx} = ξ_{xx} =ξ_{x}ξ_{x}= ξ^2_{x}## . Also we may have
##U_{xy} =ξ_{xy} =ξ_{x}ξ_{y}.## the other transforms follow in a similar manner.
Say you want to find the following Integrals
$$\int \frac{1}{(x-1)(x+2)} (dx)$$
$$\int \frac{1}{(x-1)(x^2 + 2)} (dx)$$
The easiest way to solve them will be by using partial fraction decomposition on both the given functions.
Decomposing the first function,
$$\frac{1}{(x-1)(x+2)} =...
Attempt at question No. 1:
ΔD = ∂D/∂h * Δh + ∂D/∂v * Δv
∂D/∂h = 3Eh^2/(12(1-v^2))
∂D/∂v = 2Eh^3/(12(1-v^2)^2)
Δh = +- 0,002
Δv = 0,02
h = 0,1
v = 0,3
ΔD = 3Eh^2/(12(1-v^2)) * Δh + 2Eh^3/(12(1-v^2)^2) * Δv
Because the problem asked for maximum percentage error then I decided to use the...
Hi Pfs
Partial tracing maps what occurs in a big Hilbert space toward a smaller one. We have to use it when degrees of freedom are physically unobservable or when we have only a coarse grained view of the environment. it is like in Flatland , where the two dimensional inhabitants has no access...
Hello!
Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...
Hello! Consider this partial differential equation
$$ zu_{xx}+x^2u_{yy}+zu_{zz}+2(y-z)u_{xz}+y^3u_x-sin(xyz)u=0 $$
Now I've got the solution and I have a few questions regarding how we get there. Now we've always done it like this.We built the matrix and then find the eigenvalues.
And here is...
Greetings!
I want to caluculate the summation of this following serie
I started by removing the 4 by
and then
and I thought of the taylor expansion of
Log(1-x)=-∑xn/n but as the 2 is not inside (-1,1) I couldn´t use it
any hint?
thank you!
Best !
Let $$y=\frac {1+3x^2}{(1+x)^2(1-x)}= \frac {A}{1-x}+\frac {B}{1+x}+\frac {C}{(1+x)^2}$$
$$⇒1+3x^2=A(1+x)^2+B(1-x^2)+C(1-x)$$
$$⇒A-B=3$$
$$2A-C=0$$
$$A+B+C=1$$
On solving the simultaneous equations, we get ##A=1##, ##B=-2## and ##C=2##
therefore we shall have,
$$y=\frac {1}{1-x}+\frac...
How is the order of a partial differential equation defined?
This is said to be first order: ##\frac{d}{d t}\left(\frac{\partial L}{\partial s_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0##
And this second order :##\frac{d}{d t}\left(\frac{\partial L}{\partial...
I am reading on this part; and i realize that i get confused with the 'lettering' used... i will use my own approach because in that way i am able to work on the pde's at ease and most importantly i understand the concept on separation of variables and therefore would not want to keep on second...
ok obviously easy but I never heard of the terminology for division
a friend sent me this screen shot so I don't know the explanation given
it seem more complicated than it needs to be
Anyway Mahalo if you are familiar with this
The first plot shows a large number of terms of Zeta(0.5 + i t) plotted end to end for t = 778948.517. The other plots are two zoomed-in regions, including one ending in a Cornu spiral. Despite all sorts of vicissitudes, the plot generally spirals outwards in a "purposeful" sort of way. It is...
Hi
For a function f ( x , t ) = 6x + g( t ) where g( t ) is an arbitrary function of t ; then is it correct to say that f ( x , t ) is not an explicit function of t ?
For the above function is it also correct that ∂f/∂t = 0 because f is not an explicit function of t ?
Thanks
$$\sum_i (\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j}\dot{q}_i)+\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j})\ddot{q}_i)+\frac{\partial}{\partial t}(\frac{\partial T}{\partial \dot{q}_j})$$
They wrote that above equation is equal to...
Intersecting the graph of the surface z=f(x,y) with the yz -plane.
This is the option I have chosen, but it's wrong. I don't understand why. x is fixed so I thought the coordinates: y and z are left.
I thought this source may be helpful...
Good day
I just want to confirm if a function f(x,y) who has directional derivatives has automatically partial derivatives (even though the function itself is not necessarly differentiable)? Can we consider that partial derivatives are special cases of directional derivatives?
Thank you in advance!
Hello, I am trying to calculate the partial derivative of a convolution. This is the expression:
##\frac{\partial}{\partial r}(x(t) * y(t, r))##
Only y in the convolution depends on r. I know this identity below for taking the derivative of a convolution with both of the functions only...
Problem: If sequence ## (a_n) ## has ##10-10## as partial limits and in addition ##\forall n \in \mathbb{N}.|a_{n+1} − a_{n} |≤ \frac{1}{n} ##, then 0 is a partial limit of ## (a_n) ##.
Proof : Suppose that ## 0 ## isn't a partial limit of ## (a_n) ##. Then there exists ## \epsilon_0 > 0 ## and...
Heisenberg equation of motion for operators are given by
i\hbar\frac{d\hat{A}}{dt}=i\hbar\frac{\partial \hat{A}}{\partial t}+[\hat{A},\hat{H}].
Almost always ##\frac{\partial \hat{A}}{\partial t}=0##. When that is not the case?
I've attached images showing my progress. I have used Maxwell relations and the definitions of ##\alpha##, ##\kappa## and ##c##, but I don't know how to continue. Can you help me?
It looks very easy at first glance. However, the variable S is a variable in the given expression. I have no clue to relate the partial derivatives to entropy and the number of particles.
Can someone explain me some studies I saw about partial reprogramming and rejuvenation?.
In Vivo Amelioration of Age-Associated Hallmarks by Partial Reprogramming - https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5679279/
Multi-omic rejuvenation of human cells by maturation phase transient...
Hi all,
I am having some problems expanding an equation with index notation. The equation is the following:
$$\frac {\partial{u_i}} {dx_j}\frac {\partial{u_i}} {dx_j} $$
I considering if summation index is done over i=1,2,3 and then over j=1,2,3 or ifit does not apply.
Any hint on this would...
Hi all, I was wondering is if the following partial derivative can be computed without a specific ##u(t,x)##
$$\partial_tu\big[(t,x-t\kappa V)\big]$$
I was thinking it can't be done, because we could have
$$u_a(t,x)=tx \Rightarrow \partial_tu\big[(t,x-t\kappa...
I have read numerous times that equilibrium vapor pressure (EVP) is a function ONLY of temperature. This at least partly makes sense to me (so I think) given energy of molecules and movement associated with such. But apparently this is not true for the partial pressures?
I once thought that...