What is Partial: Definition and 1000 Discussions

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).

View More On Wikipedia.org
  1. F

    I Partial derivatives of the function f(x,y)

    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...
  2. N

    Correct Usage of Partial Derivative Symbols in PDEs

    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...
  3. Mohammad-gl

    A Can I calculate partial density of states using tight binding?

    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?
  4. P

    Partial molar quantities

    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...
  5. G

    Calculating atmospheric partial pressure of oxygen

    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...
  6. C

    I Lacking intuition with partial derivatives

    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...
  7. MatinSAR

    Not understanding these manipulations involving Partial Derivatives

    Can someone please help me to find out what happened here ?
  8. haha0p1

    Finding partial pressure at equilibrium

    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...
  9. chwala

    Solve the given partial differential equation

    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]##
  10. yucheng

    I Partial trace and the reduced density matrix

    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...
  11. chwala

    A Solve the Partial differential equation ##U_{xy}=0##

    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...
  12. M

    Regarding the nth partial sum

    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?
  13. V

    Gaussian Elimination of Singular Matrix with partial pivoting

    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...
  14. C

    Free fall with a partial time provided

    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...
  15. C

    I Carroll GR: Tangent Space & Partial Derivatives

    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...
  16. karush

    Determine the order of differentiation for this partial differential eqn

    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
  17. tbn032

    Limits of Partial Charges in Dipoles

    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...
  18. L

    B Question about the definition of a partial derivative

    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...
  19. G

    I Understanding Covariant and Partial Derivatives in General Relativity

    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!
  20. J

    Calculating the partial derivative in polar coordinates

    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}...
  21. Haorong Wu

    Solving a partial differential equation

    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...
  22. chwala

    Proof involving ##ω(ξ,n)=u(x,y)## - Partial differential equations

    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.
  23. Physics Slayer

    A doubt in Partial fraction decomposition

    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)} =...
  24. manareus

    Estimating maximum percentage error using partial differentiation

    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...
  25. H

    A Does partial tracing make us see things that do not occur?

    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...
  26. A

    Partial fraction decomposition with Laplace transformation in ODE

    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...
  27. S

    Classification of a second order partial differential equation

    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...
  28. A

    What is the summation of this partial sequence?

    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 !
  29. chwala

    Solve the problem that involves partial fractions

    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...
  30. K

    I Definition of order of a partial differential equation

    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...
  31. chwala

    Using separation of variables in solving partial differential equations

    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...
  32. M

    MHB How to solve this partial differential equation

    What the hell is this and is it solvable?
  33. karush

    MHB 04 scaffold partial products for division

    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
  34. S

    I Broadly spiralling shape from partial sums of Zeta (0.5 + i t)

    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...
  35. D

    I Partial differentiation and explicit functions

    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
  36. Istiak

    Find that this partial differentiation is equal to 0

    $$\sum_i (\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j}\dot{q}_i)+\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j})\ddot{q}_i)+\frac{\partial}{\partial t}(\frac{\partial T}{\partial \dot{q}_j})$$ They wrote that above equation is equal to...
  37. Poetria

    Partial differential (multivariable calculus)

    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...
  38. A

    Directional derivatives vs Partial derivatives

    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!
  39. J

    I Partial Derivative of Convolution

    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...
  40. mcas

    Show that a partial molar property is an intensive property

    I started by taking a derivative: $$E = \sum_{i=1}^{\alpha} (E_i^{(p)} n_i) \ \ \ | \cdot \frac{\partial}{\partial n_i}$$ $$\frac{\partial E}{\partial n_i}=\sum_{i=1}^{\alpha} [\frac{\partial E_i^{(p)}}{\partial n_i}n_i + E_i^{(p)} \frac{\partial n_i}{\partial n_i}]$$ $$\frac{\partial...
  41. C

    I ##(a_n) ## has +10,-10 as partial limits. Then 0 is also a partial limit

    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...
  42. L

    A Heisenberg equation of motion -- Partial derivative question

    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?
  43. Like Tony Stark

    Partial derivatives of enthelpy and Maxwell relations

    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?
  44. Mayan Fung

    Relating the entropy of an ideal gas with partial derivatives

    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.
  45. F

    Partial Reprogramming and Rejuvenation

    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...
  46. S

    A Index notation and partial derivative

    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...
  47. docnet

    Help taking a partial derivative

    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...
  48. E

    How pressure affects partial pressures (and dewpoint)

    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...
Back
Top