Finding the extrema; an optimization problem you can see math class
Question: Find the extrema of \(x^2+2y^2+3z^2\) under the condition that \(3x+2y+z=-1\). Answer: From the condition, solve for \(z\). \[ z […]
Lagrangian function, which is for junior level of physics, of a sphere on a cylindrical surface
Question: There is an open cylinder of radius \(R\), which is stationary. A hollow sphere of radius \(\rho\) and mass […]
Residue theorem and integral of a complex-valued function: The climax of complex variables
Question: Find the value of \[ \int_C \frac{dz}{z^2+2iz-4} \] The contour, \(C\), is a circle that has center \(z=1\) and […]
Investigate the mechanical model of the proton (angular momentum etc.)
Question: The proton’s mass and radius are 1.67 \(\times\) 10\(^{-27}\) kg and 1.00 \(\times\) 10\(^{-15}\) m. This system is governed […]
What is the difference between intensity, power, and energy?
Question: There is a speaker emitting sound with a uniform power of 250 W. At what distance will the intensity […]
The basic of simultaneous ordinary differential equation: Easy one
Question: Solve for the following simultaneous differential equations: \[ (1): \frac{dx}{dt}+ay = e^{bt} \] \[ (2): \frac{dy}{dt}-ax = 0 […]
Normalize a wave function: Quantum mechanical harmonic oscillator
Question: The wave function for the one dimensional harmonic oscillator with the potential energy, \(\frac{1}{2}kx^2\), is given as \[ \Phi_0 […]
The integral of \(\frac{1}{1+x^6}\) using Beta and Gamma functions
Question: Find the value of integral \[ \int^{\infty}_{-\infty}\frac{dx}{x^6+1} \] Answer: The integrand is even function, so we can reduce the […]
The Newton’s equation of motion when you push two objects at once!
Question: The objects A and B have masses, 20.0 kg and 10.0 kg, respectively. They are placed on a frictionless […]
Differentiation with trigonometric and logarithmic functions
Question: Here is a function of \(x\). \[ y=\alpha \cos x + 2 – \cos x \log\frac{1+\cos x}{1-\cos x} \] \(\alpha\) […]