What is Differentials: Definition and 240 Discussions
In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by
d
y
=
f
′
(
x
)
d
x
,
{\displaystyle dy=f'(x)\,dx,}
where
f
′
(
x
)
{\displaystyle f'(x)}
is the derivative of f with respect to x, and dx is an additional real variable (so that dy is a function of x and dx). The notation is such that the equation
d
y
=
d
y
d
x
d
x
{\displaystyle dy={\frac {dy}{dx}}\,dx}
holds, where the derivative is represented in the Leibniz notation dy/dx, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes
d
f
(
x
)
=
f
′
(
x
)
d
x
.
{\displaystyle df(x)=f'(x)\,dx.}
The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. Traditionally, the variables dx and dy are considered to be very small (infinitesimal), and this interpretation is made rigorous in non-standard analysis.
In free time I start to solve differentials and integrals, I am doing fine, I just follow rules and solve the tasks.
I start solve some applied calculus tasks, but I dont really understand why for exmple second derivative represent acceleration, why first is speed, why I need to derivate...
Apologies for my lack of knowledge on the equations front, I have burnt by brain out on this and haven't the capacity to learn LaTex right now! So here's a screengrab of it:
So, This is a Matlab coursework and I am struggling to work out how best to approach solving it. What I have so far is...
Summary:: This is similar to the examples of electrical circuit state space analysis, I have been trying to find the state space equations from the following non linear first order differentials but I keep getting stuck. Any help?
A) Started off from non linear equations:
$$y' =...
Hi, PF, want to know how can I go from a certain error formula for linearization I understand, to another I do not
Error formula for linearization I understand:
If ##f''(t)## exists for all ##t## in an interval containing ##a## and ##x##, then there exists some point ##s## between ##a## and...
I am going through some proofs for Damping oscillations in relation to partial differentials. Can someone help on why the numbers are switched around after giving inequality condition? Please see the images for better clarity. The highlighted characters that gets switched around.
Thank you in...
https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/2.-partial-derivatives/part-c-lagrange-multipliers-and-constrained-differentials/session-42-constrained-differentials/MIT18_02SC_pb_42_comb.pdf
In part 3 of this example, after dy was expressed in terms of dx, can...
What I don't understand is why ##dS## is expanded in only the two differentials ##dV## and ##dT.## Why doesn't it look more like:
$$dS = \left(\dfrac{\partial S}{\partial V}\right)_{T,P,U} \ dV + \left(\dfrac{\partial S}{\partial T}\right)_{V,P,U} \ dT + \left(\dfrac{\partial S}{\partial...
Disclaimer: I am not a mathematician, I am a physicist.
The thermodynamic identity is usually expressed in the following differential form
$$
dU = TdS - PdV + \mu dN,
$$
where U , T , S , P , V , \mu and N are the internal energy, temperature, entropy, pressure, volume, chemical...
Hi,
I am an undergrad looking to purchase a good textbook on differentials for my course which I will be taking soon, and the textbook listed for the differentials course is this one (https://www.amazon.com/gp/product/1118531779/?tag=pfamazon01-20) which apparently is not very good. So can...
In Thermodynamics, I have seen that some equations are expressed in terms of inexact differentials, ##\delta##, instead of ##d##. I understand that this concept is introduced to point out that these differential forms are path-dependent, although I am not clear how they can be handled.
So, are...
image due to graph, I tried to duplicate this sin wave on desmos but was not able to.
so with sin and cos it just switches to back and forth for the derivatives so thot a this could be done just by observation but doesn't the graph move by the transformations
well anyway?
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...
I need some help with fully understanding some remarks by Browder made after the proof of...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...
I need some help with an example based only on Proposition 8.12 ... and the material in...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...
I need some further help in fully understanding some remarks by Browder made after...
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... ...
I need some further help in fully understanding a remark by Browder after Definition 8.9...
First I list what I am given:
Diameter: 46m
DR: 0.0003m (converted from the 0.03cm that was given)
DV: ?
Volume of a Sphere = 4/3 pi r^3
But volume of a half a Sphere = 2/3 pi r^3
And Radius = Diameter/2
So...dv = 2 pi (23m)^2 (0.0003m)
Using a Calculator I am given 0.3174. The question asks...
There is an ambiguity in certain texts that I want to clarify, atleast it seems ambiguous to me. When describing the differential line element in Relativity by the differential dx's, are they to be infinitesimal vectors or just infinitesimal increments. They are labeled coordinate differentials...
I fell upon such a wrting :
$$du=tan(d\theta)$$
How to integrate this ?
I didn't try numerically but I thought of expanding the tangeant in series but then should for example $$\int d\theta^2$$ be understood as a double integration ?
if ##d^3x=Jd^3X...(1) ## where ##x's## evolves with time and ##X's## are constt. and ##x_i=f(X_i)##(for ##i^{th}## coordinate) where the functional form of ##f(X_i)## changes with the time evolution of ##x_i##.
Now taking time derivative of (1) and dividing throughout by (1) it is coming ##\dot...
Please see the attached image. It's a part of a study guide for my final, but I didn't put it in the homework section because I already got the answer, I just don't know what it means.
The question has to do with the differential of an arc length. I made some drawings to see if I could make some...
I am reading the book "An Introduction to Differential Manifolds" (Springer) by Jacques Lafontaine ...
I am currently focused on Chapter 1: Differential Calculus ...
I need help in order to fully understand a remark by Lafontaine following his definition of differentials ...
Lafontaine's...
We are trying to design a new product which uses gas pressures
in various ways. We are a little foxed about partial pressures:
Imagine a small cylinder; 6in long + 1in in diameter.
The cylinder is made from steel.
The cylinder is divided into two equal compartments – A and B.
Each compartment...
A point is moving on the graph of
3x^2 + 4y^3 = xyWhen the point is at P = (1/7, 1/7)
its y-coordinate is increasing at a speed of 3
units per second.
What is the speed of the x-coordinate at
that time and in which direction is the xcoordinate
moving?
I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ...
I am currently focused on Chapter 2: Derivation ... ...
I need help with an aspect of Kantorovitz's Example 4 on page 66 ...
Kantorovitz's Example 4 on page 66 reads as follows:In the above example, Kantorovitz...
I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ...
I am currently focused on Chapter 2: Derivation ... ...
I need help with an aspect of Kantorovitz's Example 4 on page 66 ...
Kantorovitz's Example 4 on page 66 reads as follows:
In the above example, Kantorovitz...
So I've seen in several lectures and explanations the idea that when you have an equation containing a relation between certain expressions ##x## and ##y##, if the expression ##x## approaches 0 (and ##y## is scaled down accordingly) then any power of that expression bigger than 2 (##x^n## where...
one side of a right triangle is known to be 12 cm long and the opposite angle is measured as 30°, with a possible error of ±1°.
(a) Use differentials to estimate the error in computing the length of the hypotenuse.
Homework Statement
I am working on this problem and having difficulty getting the required answer. It is the exact problem as here , but I’m still not getting it.
BTW this is problem 10, Section 4, Chapter 4 Partial Differentiation from M. Boas’s book Mathematical Methods in the Physical...
I know how to prove the quotient rule by using the definition of a derivative using limits (Newton's style). I just saw a proof of the product rule using Leibniz's concept of differentials on Wikipedia. https://en.wikipedia.org/wiki/Product_rule#Discovery
Does anyone know of a Leibniz-style...
Hi! Can someone give me an example of a function ##f(x,y)## for which the mixed partial differentials are not equal, i.e. $$\frac{\partial^2 f}{\partial x \partial y} \neq \frac{\partial^2f}{\partial y \partial x}$$
It says in Boas that these mixed differentials are equal only if the first and...
I was just wondering is there any way to calculate what the airspeed would be, inside of a pipe with a known diameter and length, that lay with each end exposed to a known, different pressue than the other.
Was wondering if such a set up would be feasible for generating clean electricity by...
Find the particular solution determined by the given condition:
ds/dt = 14t^2+3t -3; s=124 when t = 0
Could someone please point me in the right direction to start this problem? I'm not even sure if I titled this post correctly. :P
Thank you!
Homework Statement
A coat of paint of thickness 0.02 in is applied to the faces of a cube whose edge is 10 in, thereby producing a slightly larger cube. Use differentials to find approximately the number of cubic inches of paint used. Also find the exact amount used by computing volumes before...
Homework Statement
Consider a closed rectangular box with a square base of side 3cm and height 5cm. If the side is measured with an error at most 0.02cm and the height is measured with an error at most 0.01cm, use differentials to estimate the maximum possible error in computing the volume of...
I know its a ridiculous question, but I'm signing up for classes and I wanted to take differential equations and its currently filled up(for now, people drop all the time at my uni) but saw partial differential equations. I assume you need differential equations before partial differential...
Homework Statement
2 adjacent sides of a parallelogram measure 15ft and 10ft w/ max errors of ±0.1ft
angle is 45° w/ max error of ±0.5°
What is the maximum error in the calculated value of the area or the parallelogram?
Homework Equations
A = area = xysinθThe Attempt at a Solution
x = 15ft...
Why can't (1.39) be put in the form of an exact differential? Seems like I could and the solution to the first equation is
##x-a\phi\sin\theta=c##, where ##c## is an arbitrary constant.
Let ##x-a\phi\sin\theta## be ##h##.
By considering ##dh=\frac{\partial h}{\partial x}dx+\frac{\partial...
Greetings all,
In my quest to imagine the ultimate sportscar, I began to wonder about the following:
Certain cars are All Wheel Driven (AWD) and they do so via a Center differential delivering power to both axles. Amongst those cars, some have fixed Torque Bias/Split Ratio, which doesn't alter...
Homework Statement
We're told to show the attached equation, (partial P/partial V) at constant S = that big mess.
Homework Equations
dE=TdS - PdV
The Attempt at a Solution
lol just went in circles with the commuter and permuted rules. I have no clue how to end up where I'm supposed to be.
Greetings all,
In Hartle's Gravitation, it seems like a given the way he manipulates the "differentials" when he gives a line element. The thing is that it works, and I know it's not mathematically rigorous, but first of all, I'm wondering when treating the differentials in the line elements in...
I've just done a derivation and had to use the following
u_{b}u^{c}\partial_{c}\rho = u_{b}\frac{dx^{c}}{d\tau}\frac{\partial\rho}{\partial x^{c}} = u_{b}\frac{d\rho}{d\tau}
We've done this cancellation a lot during my GR course, but I'm not clear exactly when/why this is possible.
EDIT: is...
Homework Statement
Express the quantity ∂2/∂x2+∂2/∂y2 in polar coordinates.
Homework Equations
x=ρcosφ
y=ρsinφ
ρ=sqrt(x2+y2)
The Attempt at a Solution
This is my first post, so I apologize for any weird looking equations, etc. I know that this is not a difficult problem, but I just cannot...
I need to get a few things straight about the integration operation (as an intro calc student). I understand that integration is a process that takes a function and returns its antiderivative. We can think of it as an operator, where ##\displaystyle \int...dx## is kind of like an opening and a...
I am currently taking Calculus 1 and we covered Linearization and Differentials. The title of the section in my textbook is called "Linear Approximations and Differentials," in the book by James Stewart. The book and lecture in this section made absolutely no sense to me. Like, I was COMPLETELY...
Hello folks,
I am having difficulty comprehending some material in my fluid dynamics course. This is not a homework question, just something missing in my understanding.
When proving the "Pressure Field Equation," (something I am not yet able to do) there is a series of steps my instructor...