Partial differentiation Definition and 9 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).
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...
$$\sum_i (\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j}\dot{q}_i)+\frac{\partial}{\partial q_i}(\frac{\partial T}{\partial q_j})\ddot{q}_i)+\frac{\partial}{\partial t}(\frac{\partial T}{\partial \dot{q}_j})$$
They wrote that above equation is equal to...
In Section 7.6 - Equivalence of Lagrange's and Newton's Equations in the Classical Dynamics of Particles and Systems book by Thornton and Marion, pages 255 and 256, introduces the following transformation from the xi-coordinates to the generalized coordinates qj in Equation (7.99):
My...
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}...
I am working my way through a textbook, and whenever this equation is solved (integrated), the answer is given as:
u = f(x) + f(y)
I don't understand it. If I integrate it once (with respect to y, say), then I obtain:
∂u/∂x = f(x) -----eq.1
If I integrate again (this time with respect to...
Hello guys,
I'm doing my physics coursework on kepler's third law and I'm finding the minimum mass and semi-major axis of a unknown planet. I have the following data:
Stellar mass Mstar = 1.31 ± 0.05 Msun
Orbital period P = 2.243752 ± 0.00005 days
Radial velocity semi-amplitude: V = 993.0 ±...
Hi, in the above why is the left-hand side simple differentiation, i.e V is only function of t but in the right it is function of t, x, y, and z. It is very strange that one side is different than the other. Would you like to explain it?
Thank you.