One thing that confuses me is the physical speed and sound speed. The lattice sound speed cs=1/sqrt{3} corresponds to the physical sound speed for isothermal flow (sqt{RT}). Why isn't the physical speed (e.g. inlet speed up of lid cavity) converted and use accoringly?
$$c_p=\sqrt{RT}≈330m/s...
$$V_{HE}=\sqrt{\frac{\mu}{a}}$$
What is the rationale for this formula when we can determine the change in velocity from Earth's orbit to transfer orbit using the vis-viva equation? Likewise, what is the use of defining the radius for the sphere of influence for interplanetary transfer...
Homework Statement
Water is supplied at 4.50 m3/s and 415 kPa (abs) to a hydraulic turbine through a 1.0-m inside diameter inlet pipe as indicated in the figure. The turbine discharge pipe has a 1.2-m inside diameter. The static pressure at section (2), 3 m below the turbine inlet, is 25 cm Hg...
How do I then ensure that the property of metric tensor is expressed in the equation then? Do I leave the metric tensor and the contravariant vector as separate then?
My mistake because I did assume that ##g^{\mu \nu} = diag(1, 1, 1)## which shouldn't be the case.
$$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial x^{\nu}} \\ =\frac{-9}{(\eta_{\mu \kappa}q^\mu x^\kappa)^4}[\eta_{\mu...
When you mean check answer, do you mean something like this?
E.g.
$$q=(a\ \ \ \ \ b)\\ x=(x_1\ \ \ \ \ x_2)$$
$$\partial_\nu \frac{3}{(ax_1+bx_2)^{3}}=\begin{pmatrix}-9a(ax_1+bx_2)^{-4}\\-9b(ax_1+bx_2)^{-4}\end{pmatrix} \\=\frac{-9}{(ax_1+bx_2)^{4}}q_\nu \\=\frac{-9}{(q.x)^{4}}q_\nu$$
And how...
No unless is curved spacetime. Correcting for the ημν and v,
$$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial x^{\nu}} \\ =\frac{-9}{(\eta^{\mu \kappa}q_\mu x_\kappa)^4}[\eta^{\mu...
I don't get why my answer is not right. Even when I reduced it to a normal derivative of x, I will get the same result.
$$\frac{\partial}{\partial x}\frac{3}{(qx)^3}=\frac{-9}{(q x)^4}q$$
Homework Statement
Solve this, $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}$$
where q is a constant vector.
Homework EquationsThe Attempt at a Solution
$$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial x^{\nu}}...
In Einstein summation convention, the summation occurs for upper indices and its repeated but lower indices. However I have some confusion
1) $${\displaystyle v=v^{i}e_{i}={\begin{bmatrix}e_{1}&e_{2}&\cdots &e_{n}\end{bmatrix}}{\begin{bmatrix}v^{1}\\v^{2}\\\vdots \\v^{n}\end{bmatrix}},\ \qquad...