Show C_v=T(dS/dT)_v
(Lowercase v's are the subscripts)
use the chain rule to find dz/ds and dz/dt. z=arcsin(x-y), x=s^2+t^2, y=2-6st. dz/ds=? dz/dt=?
dn (a) Show that L[i" f(t)] = (-1)" (t) for any positive integer n 2 1 dsn a d K(s, t)f(t) dt / ) est = tne-8t and assume that K(s, t)f(t) dt. Hint: (-1)" as ds (b) Use the above formula to compute L[t? cost]. dn (a) Show that L[i" f(t)] = (-1)" (t) for any positive integer n 2 1 dsn a d K(s, t)f(t) dt / ) est = tne-8t and assume that K(s, t)f(t) dt. Hint:...
Case wTIEle u - D. 3.53. Show that the Frenet-Serret formulas can be written dT = wx T, ds dB = wx B. ds in the form Also, determine w. =wxN, ds Case wTIEle u - D. 3.53. Show that the Frenet-Serret formulas can be written dT = wx T, ds dB = wx B. ds in the form Also, determine w. =wxN, ds
For s > 0 define the gamma function I (s) by T () = [co-dt. Show that I (8) extends to an analytic function in the half-plane 20 = {ZEC: Rez >0}, and that the above formula continues to hold there. Hint: Show that S T. (s) ds = 0 for every triangle T in C where I (8) = le-+48-1dt for S E C and 0 <€ < 1.
2. Heat equations. Prove the following equations for T dS, where we are assuming that the number of parti- cles N is not changing. a. A first equation, Та dV КТ T dS Ncv dT b. A second equation, T dS Ncp dT- TVa dP.
dS/dt = vR - ?SI dI/dt = ?SI – ?I dR/dt = ?I - vR. Consider a SIRS model. Modify the above equations to include vital dynamics (births and deaths from causes other than infection) such that all progeny of infected individuals are born infected. Be sure to clearly define any new variables or parameters. Explain the changes you have made.
Find the derivative of the function. s={Intl ds 11 dt
(6). The quantities x(t) and y(t) satisfy the simultaneous equations dt dt dx dt where x(0)-y(0)-ay (0)-0, and ax (0)-λ. Here n, μ, and λ are all positive real numbers. This problem involves Laplace transforms, has three parts, and is continued on the next page. You must use Laplace transforms where instructed to receive credit for your solution (a). Define the Laplace Transforms X(s) -|e"x(t)dt and Y(s) -e-"y(t)dt Laplace Transform the differential equations for x(t) and y(t) above, and incorporate...
Direct solve the differentiation S* e-J* P(e)dt ds
(a) Show that dB/ds is perpendicular to B 0 BI= 1-B-B- (b) Show that dB/ds is parpendicular to T B Tx N T' N' [(T'x N)(T * N Irtt) rit) [(TTT (Tx N rt) (c) Doduce from parts (a) and (b) that dB/ds -risN for some numbar ria called the torsion of the curve, (The torsion meacures the dearoe of twisting of a curve.) TIN. P LT and BI N. So B T and N torm -Select set of vectors...