Recent content by TimeRip496

$$V_{HE}=\sqrt{\frac{\mu}{a}}$$ 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...

Okay thanks! Was really confused by the terms given

Shouldn't 25cm Hg vacuum means 25cm Hg above vacuum pressure which is zero? Why is 25cm Hg vacuum equivalent to 25cm Hg below atm?

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...

Okay thanks for the help!

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. $$\frac{\partial}{\partial x}\frac{3}{(qx)^3}=\frac{-9}{(q x)^4}q$$

Sorry about the inner product. I just want confirm the validity of my answer. $$\frac{-9}{(q.x)^4}q_{\nu}$$

Thanks a lot!

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...

I am sorry I don't quite get the direction of the flux part. For the "flux through a section of a closed surface is positive if the vector from the mass to the section of the surface points outward. The flux is negative if it points inward.", how are the direction for the flux determined? How do...