I got a quick question regarding the potential energy of a fluid (see picture). If I want to express the potential energy of a fluid (with a given volume) above the reference line x, using nothing but the given variables/constants, would this be the correct way to do so?
picture
Of course. It makes a lot more sense to look at the substitution in an integration. Thanks for the help. Will try part 4) and post the result later.
Edit: What do you mean by "yours will integrate to 1/2"? Do I need to change the limits according to z(y)?
I gave part 3 another try using the so called "change of variable" technique. Although, instead of finding and taking the derivative of the CDF I used the definition of normal distribution (pdf) and just substituted z. I set z=g(y)=\sqrt{y} \Rightarrow \frac{dg}{dy}=\frac{1}{2\sqrt{y}}...
Alright, according to the definition, the distribution should be
$$
\begin{align}
F_Z(u) = 1-(1-F_X(u))(1-F_Y(u)) = 1-P(X>u)P(Y>u)
\end{align}
$$
where F_Y(u)=\sum_{k:k\leq u}q^k and F_X(u)=\sum_{k:k\leq u}p^k.
Homework Statement
Two independent series of experiments are performed. The probability of a positive result (independent of each other) in the respective series are given by p and q. Let X and Y be be the amount of experiments before the first negative result occur in the respective series...
Yes, I'm aware of this. Solving this with ode45 should yield the x and y vectors that I want to plot. I'm just not sure how to define the function itself since it depends on B which in turn depends on y.
$$
\begin{align}
B(y) = B_{0}e^{-y/y_{0}}
\end{align}
$$
I'm working on a little project where I want to plot the motion of a projectile with air resistance. The air resistance can be assumed to be proportional to the velocity squared.
F_{f}=-Bv^{2}
F_{f,x}=F_{f}\frac{v_{x}}{v}, \ \ F_{f,y}=F_{f}\frac{v_{y}}{v}
where B depends on the height...
Plugging in the values for the partial derivatives gives
\frac{\partial}{\partial x} \left[4\frac{\partial F}{\partial v}+F(u,v) \right] = 0 = \frac{\partial}{\partial x}g(y).
Does that mean the general solution is
F(u,v) = g(y)-4\frac{\partial F}{\partial v}
where g(y) and...
Homework Statement
Find the general solution f = f(x,y) of class C2 to the partial differential equation
\frac{\partial^2 f}{\partial x^2}+4\frac{\partial^2 f}{\partial x \partial y}+\frac{\partial f}{\partial x}=0
by introducing the new variables u = 4x - y, v = y.
Homework Equations...
I am soon to apply for university and I am not quite sure what I want to study. Theoretical physics and astrophysics/astronomy/cosmology all seem like very interesting fields of physics to me. How do I go about when I choose my career path? Which one of the fields has been more prospering than...