What is Partial derivative: Definition and 373 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).
I’m having a little confusion about part b of this question as to why I am allowed to use the limit definition of a partial derivative.
Here’s what I think:
I know that y^3/(x^2+y^2) is undefined at the origin but it does approach 0 when it GETS CLOSE to the origin. So technically defining...
for the boundary conditions for this problem I understand that Electric field and Electric potential will be continuous on the boundaries.
I also know that I can set the reference point for Electric potential, wherever it is convenient. This gives me the fifth boundary condition. I am confused...
Hi there!
I would like to know if the following simplification is correct or not:
Let A be a function of x, y, and z
$$\frac{\partial^2A}{\partial x^2}+\frac{\partial^2A}{\partial y^2}$$
$$=\ \frac{\partial^2A\partial y^2+\partial^2A\partial x^2}{\partial x^2\partial y^2}$$...
I just wanted to know if my solution to part (b) is correct. Here's what I did:
I took the partial derivative with respect to x and y, which gave me respectively.
Then I computed the partial derivatives at (-3,4) which gave me 3/125 for partial derivative wrt x and -4/125 for partial derivative...
The answer to this problem is
However, I am confused how this relates to the question.
My working is,
##V = cT^2 - bpT##
##\frac{dV}{dT} = 2cT - bp## (I take the partial derivative of volume with respect to temperature to get the isobaric expansion coefficient)
##\frac{dV}{dP} = 0## (I take...
Given a function F(x,y)=A*x*x*y, calculate dF(x,y)/d(1/x), to calculate this derivative I make a change of variable, let u=1/x, then the function becomes F(u,y)=A*(1/u*u)*y, calculating the derivative with respect to u, we have dF/du=-2*A*y*(1/(u*u *u)) replacing we have dF/d(1/x)=-2*A*x*x*x*y...
To find a shock wave, do we always solve the equation ##x_{\xi}=0##? The PDEs I consider are of the form ##u_t + g(u) u_x = f(u)##, with initial condition ##u(x,0) = h(x)##. I have been looking at the solutions for problems in my homework sheet but this method was used with no explanation.
Why...
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 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}...
Given that the partial derivatives of a function ##f(x,y)## exist and are continuous, how can we prove that the following limit
$$\lim_{h\to 0}\frac{f(x+hv_x,y+hv_y)-f(x,y+hv_y)}{h}=v_x\frac{\partial f}{\partial x}(x,y)$$
I can understand why the factor ##v_x## (which is viewed as a constant )...
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...
Hey guys! I'm currently struggling with a specific thermodynamics problem.
I'm given the entropy of a system (where ##A## is a constant with fitting physical units): $$S(U,V,N)=A(UVN)^{1/3}$$I'm asked to calculate the specific heat capacity at constant pressure ##C_p## and at constant volume...
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?
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
$$u_a(t,x)=tx \Rightarrow \partial_tu\big[(t,x-t\kappa...
So I start by isolating v
the speed here would be the square root of the partial t derivative divided by the sum of the partial x and y derivatives.
the amplitude, phi and the cos portion of the partial derivatives would all cancel out.
What I am left with is the sqrt(43.1 / ( 2.5 + 3.7 ) =...
Divergence & curl are written as the dot/cross product of a gradient.
If we take the dot product or cross product of a gradient, we have to multiply a function by a partial derivative operator.
is multiplication by a partial derivative operator allowed? Or is this just an abuse of notation
I notice that ##pv=t## looks like the ideal gas law but with ##T## in units of energy. I know that ##pV = \text{constant}## means that the pressure of a gas decreases as you expand it (Boyle-Mariotte's law), explaining for instance how we breathe.
I guess I could put everything in words even...
Why the summation of the following function will be canceled out when we take the partial derivative with respect to the x_i?
Notice that x_i is the sub of (i), which is the same lower limit of the summation! Can someone, please explain in details?
I am not sure how to determine the sign of this derivatives.
(a) first we can pass a plane by (1,2) parallel to XZ (y fixed) and see how the curve belongs to the plane will vary with x, but what about the next partial derivative, with respect to y?
Ref. 'Core Principles of Special and General Relativity' by Luscombe. Apologies in advance for the super-long question, but it's necessary to show my thought process.
Let ##\gamma:I\to M## be a smooth curve from an open interval ##I\subset\mathbb{R}## to a manifold ##M##, and let...
It's a detail, but annoying to me: ##{\partial u\over \partial x} = {\partial \phi \over \partial x} \;+ ...##
$${\partial u\over \partial x} = {\partial \phi \over \partial x} \;+ ...$$
How do I move up ##\partial u## a little bit so it aligns with ##\partial \phi## ?
I research about coordinate systems and I found the following informations about transformation.
Now, if I replace arctan (x/y) (according to the picture above) to φ, I think I can solve. But if I can do this, then what will be replaced to ψ? I mean, I know just taking partial derative about...
While working at home during the COVID-19 pandemic I've taken to seeing if I can still do math from undergrad (something I do once in a while to at least pretend my life isn't dominated by excel). So to that I've been reviewing partial derivatives (which I haven't really thought about in a good...
Hi guys,
suppose we have a function ##C(x, y)## into the real numbers. Suppose also that ##y=y(x)##, i.e. ##y## is a function of ##x##.
Now in my script, I have a term ##\nabla_x C(x_0, y(x_0)) ##. From my point of view, this means that you take the partial derivative of ##C(x,y)## with...
My attempt
I calculated the partial derivatives of n wrt P and T. They are given below.
##\frac {\partial n}{\partial P} = \frac{nb -1}{\left(2an-Pb-3abn^2-kT\right )}##
##\frac {\partial n}{\partial T}= \frac {nk}{\left(2an-Pb-3abn^2-kT \right ) }##
I know that if the partial derivative is...
The function should use (r,z,t) variables
The domain is (0,H)
Since U is not dependent on angle, then theta can be ignored in the expression for Laplacian in cylindrical coordinates(?)
I apologise for the length of this question. It is probably possible to answer it by reading the first few lines. I fear I have made a childish error:
I am working on the geodesic equation for the surface of a sphere. While doing so I come across the partial derivative
\begin{align}...
I have this function, and I want to take the derivative. It includes a unit step function where the input changes with time. I am having a hard time taking the derivative because the derivative of the unit step is infinity. Can anyone help me?
##S(t) = \sum_{j=1}^N I(R_j(t)) a_j\\
I(R_j) =...
If given a function ##u(x,y) v(x,y)## then is it correct to write ##\frac{\partial }{\partial x}u(x,y)v(x,y)=\frac{u(x+dx,y)v(x+dx,y)-u(x,y)v(x,y)}{dx}##??
It seems that the way to combine the information given is
z = f ( g ( (3r^3 - s^2), (re^s) ) )
we know that the multi-variable chain rule is
(dz/dr) = (dz/dx)* dx/dr + (dz/dy)*dy/dr
and
(dz/ds) = (dz/dx)* dx/ds + (dz/dy)*dy/ds
---(Parentheses indicate partial derivative)
other perhaps...
How do I interpret geometrically the partial derivative in respect to a constant of a function such as ##\frac{ \partial}{\partial c} (acos(x) + be^x + c)^2##?
I had already calculated the first partial derivative to equal the following:
$$\frac{\partial y}{\partial t} = \frac{\partial v}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial v}{\partial t}$$
Now the second partial derivative I can use the chain rule to do and get to...
How can I calculate ∂/∂t(∫01 f(x,t,H(x-t)*a)dt), where a is a constant, H(x) is the Heaviside step function, and f is
I know it must have something to do with distributions and the derivative of the Heaviside function which is ∂/∂t(H(t))=δ(x)... but I don't understand how can I work with the...
Homework Statement
I'm given a gas equation, ##PV = -RT e^{x/VRT}##, where ##x## and ##R## are constants. I'm told to find ##\Big(\frac{\partial P}{\partial V}\Big)_T##. I'm not sure what that subscript ##T## means?
Homework Equations
##PV = -RT e^{x/VRT}##
Thanks a lot in advance.
1. The problem statement, all variables, and given/known data
Given is a second order partial differential equation $$u_{xx} + 2u_{xy} + u_{yy}=0$$ which should be solved with change of variables, namely ##t = x## and ##z = x-y##.
Homework Equations
Chain rule $$\frac{dz}{dx} = \frac{dz}{dy}...
Homework Statement
Show that the displacement D(x,t) = ln(ax+bt), where a and b are constants, is a solution to the wave function.
Homework Equations
I'm not sure which one to use:
D(x,t) = Asin(kx+ωt+φ)
∂2D/∂t2 = v2⋅∂2D/∂x2
The Attempt at a Solution
I'm completely lost on where to start...
Homework Statement
Given
$$L = \left(\nabla\phi + \dot{\textbf{A}}\right)^2 ,$$
how do you calculate $$\frac{\partial}{\partial x}\left(\frac{\partial L}{\partial(\partial\phi / \partial x)}\right)?$$
Homework Equations
By summing over the x, y, and z derivatives, the answer is supposed to...
Hi all.
Suppose I have the ideal gas law $$P=\frac{RT}{v}$$If I'm asked about the partial derivative of P with respect to molar energy ##u##, I may think "derivative of P keeping other quantities (whatever those are) constant", so from the formula above I get $$\frac{\partial P}{\partial...