I have a vector field which is originallly written as $$ \mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$
(##r## is the distance from origin, ##\phi## is azimuthal angle and...
You basically just take the second derivative of the given function and multiply it by the original then multiple everything by m. I just don’t understand how the second derivative would be negative.
I have attached a word document demonstrating the working out cos i was too lazy to learn how Latex primer works and writing it like I did above would've been too hard too read. I tried to make it as understandable as possible, presenting fractions as
' a ' instead of ' a / b ' .
------
b
Hi all,
I have ran into some mathematical confusion when studying the aforementioned topic. The expression for retarded time is given as
$$t_R = t - R/c$$
##R = | \vec{r} - \vec{r'} |##, where ##\vec{r}## represents the point of evaluation and ##\vec{r'}## represents the source position. I...
The building of theoretical mechanics can be constructed using only the first and the second derivatives (those of coordinates in case of kinematics: velocity and acceleration and those of energy in case of dynamics: force and gradient thereof). It is obviously unavoidable if one wants to deal...
Homework Statement
(a) Consider the line integral I = The integral of Fdr along the curve C
i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C?
ii) What is the value of I if the vector field F is is a unit vector...
Homework Statement
Homework Equations
The Attempt at a Solution
I tried to find the slope of the tangent line, but this gave me 3.66 and the answer is 3.8 how do I find this?
Hi, friends! Under particular conditions on ##\phi:\mathbb{R}^3\times\mathbb{R}\to\mathbb{R}## - I think, as said here, that it is sufficient that ##\phi\in C_c^1(\mathbb{R}^4)##: please correct me if I am wrong - the following equality holds$$\frac{\partial}{\partial r_k}\int_{\mathbb{R}^3}...
Homework Statement
Evaluate the derivative of the following function:
f(w)= cos(sin^(-1)2w)
Homework Equations
Chain Rule
The Attempt at a Solution
I did just as the chain rule says where
F'(w)= -[2sin(sin^(-1)2w)]/[sqrt(1-4w^(2))
but the book gave the answer as F'(w)=(-4w)/sqrt(1-4w^(2))...
For implicit differentiation, is dy/dx of x2+y2 = 50 the same as y2 = 50 - x2 ?
From what I can take it, it'd be a no since.
For x2+y2 = 50,
d/dx (x2+y2) = d/dx (50) --- will eventually be ---> dy/dx = -x/y
Where,
y2 = 50 - x2
y = sqrt(50 - x2)
dy/dx = .5(-x2+50)-.5*(-2x)
Homework Statement
This is my 'carrying out a practical investigation' assignment for Maths. I've attached the coursework (what I've wrote up to now) and my main concern is whether I've got the right differential equation to find 3 new velocity values throughout the pendulum trajectory...
I've been thinking about it since yesterday and have noticed this pattern:
We have, the first order derivative of a function ##f(x)## is:
$$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} .......(1)$$
The second order derivative of the same function is:
$$f''(x)=\lim_{h\rightarrow...
Derivatives in first year calculus
Gateaux Derivatives
Frechet Derivatives
Covariant Derivatives
Lie Derivatives
Exterior Derivatives
Material Derivatives
So, I learn about Gateaux and Frechet when studying calculus of variations
I learn about Covariant, Lie and Exterior when studying calculus...
Let's talk about the function ##f(x)=x^n##.
It's derivative of ##k^{th}## order can be expressed by the formula:
$$\frac{d^k}{dx^k}=\frac{n!}{(n-k)!}x^{n-k}$$
Similarly, the ##k^{th}## integral (integral operator applied ##k## times) can be expressed as:
$$\frac{n!}{(n+k)!}x^{n+k}$$
According...
The average angle made by a curve ##f(x)## between ##x=a## and ##x=b## is:
$$\alpha=\frac{\int_a^b\tan^{-1}{(f'(x))}}{b-a}$$
I don't think there should be any questions on that. Since ##f'(x)## is the value of ##\tan{\theta}## at every point, so ##tan^{-1}{(f'(x))}##, should be the angle made by...
I understand the conditions for the existence of the inverse Laplace transforms are
$$\lim_{s\to\infty}F(s) = 0$$
and
$$
\lim_{s\to\infty}(sF(s))<\infty.
$$
I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as
$$F(s) =\begin{cases} 1-s...
Consider the function:
$$F(s) =\begin{cases} A \exp(-as) &\text{ if }0\le s\le s_c \text{ and}\\
B \exp(-bs) &\text{ if } s>s_c
\end{cases}$$
The parameter s_c is chosen such that the function is continuous on [0,Inf).
I'm trying to come up with a (unique, not piecewise) Maclaurin series...
OK, I admit: this will be the most idiotic question I have ever asked (maybe: there could be more)
So, I am aware of the differential calculus (derivatives) and the integral calculus (integrals).
And separate from that, there is the first fundamental theorem (FFT) of the calculus which relates...
A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant.
Two ratios are proportional if they change equally and are related by a constant of proportionality? Not sure about this definition, but please correct it if you can...
This is (should be) a simple question, but I'm lost on a negative sign.
So you have ##D_m V_n = \partial_m V_n - \Gamma_{mn}^t V_t## with D_m the covariant derivative.
When trying to deduce the rule for a contravariant vector, however, apparently you end up with a plus sign on the gamma, and I'm...
Homework Statement
r(t) is the position of a particle in the xy-plane at time t. Find an equation in x and y whose graph is the path of the particle. Then find the particle’s velocity and acceleration vectors at the given value of t.
Homework Equations
First derivative = velocity...
I understand that the derivative of a function ##f## at a point ##x=x_{0}## is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where ##\Delta x## is a small change in the argument ##x## as we "move" from ##x=x_{0}## to a neighbouring...
Homework Statement
Homework Equations
The Quotient rule for calculating the derivative.
The Attempt at a Solution
The derivative f'(x) = (x+5)-(x+3) / (x+5)^2
I tried a previous similar problem but failed as I didn't and still don't know what '' means.
Question: Two bikers leave a diner at the same time. Biker Slim rides at 85kmh [N] and Biker Haug rides at 120kmh [NE]. How fast is the distance between them changing 40 minutes after they left?
I suggest looking at my photos of the triangles and such, as explaining it over text can be a bit...
Homework Statement
A ski jumper leaves the ski track moving in the horizontal direction with a speed of 25.0 m/s as shown in Figure 4.14. The landing incline below her falls off with a slope of 35.0°. Where does she land on the incline? I've attached an image of the problem, my work is below...
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...