calculation – Source of the formula for the divergence of the vector function in spherical coordinates

I want to calculate the divergence of the vector function. $ F (r, theta, phi) = (r ^ 2, rsin phi, 0) $. I assumed that to do this I could calculate the divergence in spherical coordinates, which would be: $ nabla F (r, theta, phi) = 2r $.

However, my textbook shows the result in Cartesian coordinates such as: $$ nabla F = frac {1} {r ^ 2sin theta} ( frac { delta} { delta r} (r ^ 2sin theta F_r) + frac { delta} { delta theta } (r sin phi F_ theta) +
frac { delta} { delta phi} (rF_ phi)) $$

I've also seen that it applies here, but they don't really give any context about the origin of this formula and it doesn't appear in any of my textbooks.

Can anyone explain / point out a good derivation of this formula? Also, why don't we take the divergent in the same coordinate system as the vector function?

Calculation – Solve complex equations $ z ^ 3 = left ( sqrt3 + i right) ^ 9 $

Solve complex equations $ z ^ 3 = left ( sqrt3 + i right) ^ 9 $

$ | z | = sqrt {3 + 1} = 2 \ sin phi = frac {1} {2} \ cos phi = frac { sqrt3} {2}, phi = frac { pi } {6} \ z_0 = sqrt (3) {2} left ( cos frac { frac { pi} {6}} {3} + i sin frac { frac { pi} {6}} {3} right) = sqrt (3) 2 left ( cos frac { pi} {18} + i sin frac { pi} {18} right PS
Wolfram says $ z_0 = 8i $ What did I do wrong?

integration – Derived from $ int_c ^ { sqrt {x}} 1dt $ – Application of the fundamental theorem of calculation

I am trying to determine the $ frac {d} {dx} int_c ^ { sqrt {x}} 1dt $.

I know that the fundamental theorem states the following,
$ frac {d} {dx} int_c ^ {{x}} f (t) dt = f (t) $

However, in this case, the function is a constant, generally if our upper limit of integration is a function in itself, we end up taking the derivative of the function by applying the chain rule as follows,
$ frac {d} {dx} int_c ^ {{g (x)}} f (t) dt = f (g (x)) cdot g & # 39; (x) $

However, in this case, since the function is a constant, can we even apply the chain rule? I think the answer should be the following,

$ frac {d} {dx} int_c ^ { sqrt {x}} 1dt = 1 $

But I'm pretty sure the correct answer is
$ frac {d} {dx} int_c ^ { sqrt {x}} 1dt = frac {1} {2 sqrt {x}} $

Could someone explain to me why my answer is incorrect?

lambda calculation – Reduction of the normal order – is the left-most order the left-most order just left?

This is a quick question. I have been reading about Lambda's calculation, and I see the normal order as "the outermost leftmost first," and the application order described as "the leftmost leftmost."

I think This makes sense to me, but I just want to make sure I understand. Is it the case that the first application in normal order will always be the leftmost lambda symbol within the lambda chain? (This assumes that there is something to apply to this lambda. If not, we would simply move on to the next one, from left to right).

calculation – Wind speed – from feet to mph

The wind speed S in mph of a tornado at a distance d feet from its center defined by $ S (a, d, V) = frac {aV} {0.51d ^ 2} $where a = 0.5 and V is the approximate volume of the tornado, in cubic feet. About the wind speed 30 feet from the center of a tornado when its volume is 12,000 cubic feet? mph (The wind speed 30 feet from the tornado is? mph).

My job:

$ S (a, d, V) = frac {aV} {0.51d ^ 2} $
$ S (a, d, V) = frac {0.5 (12000)} {0.51 (30 ^ 2)} $

$ S (a, d, V) = 13,071 feet $

The answer is in mph. How to proceed with this? Do we need to convert from feet to miles by volume and distance for the response in mph or in the final response conversion to happen?

Could you help me please?

calculation and analysis: accelerate integration with Max / Min or by parts

Integrate around a convex combination of two functions that contain Min or Max or Piecewise It doesn't end in 5 min. Conversely, Integrate it ends in one or two seconds when the argument is a linear function, and similarly when the argument contains a function with Min or Max or Piecewise, and similarly when the argument contains a convex combination of linear functions.
In the following example, the integrals of d, da, d1, d2 take <2s and the integrals of d1a, d2a not finish in 5 min.

Am I making a simple mistake or how to accelerate similar integrals of convex combinations of Piecewise functions?

MWE:

Clear(d, d1, d2, da, d1a, d2a, cdf, pdf, cdf1, pdf1, cdf2, pdf2, s,
vi, vj, vlo, mui, muj, pi, pj, pis, pjs)
$Assumptions = 
  Flatten@{Thread(0 < {s, pi, pj, pis, pjs, vlo, vi, vj}), s < 1};
cdf(v_) = v - vlo; pdf(v_) = 1;
cdf1(v_) = Max(0, Min(1, v - vlo)); pdf1(v_) = 1;
cdf2(v_) = 
 Piecewise({{0, v < vlo}, {v - vlo, vlo <= v <= vlo + 1}, {1, 
    v > vlo + 1}}); 
pdf2(v_) = 
 Piecewise({{0, v < vlo || v > vlo + 1}, {1, vlo <= v <= vlo + 1}});
d(pi_, pj_, s_, pis_) = 
  Integrate((1 - cdf(Max(pi, vj - pj + pi - s)))*pdf(vj), {vj, vlo, 
    vlo + 1});
d(0.2, 0.3, 0.1, 0.25) /. {vlo -> 0.1, mui -> 0.4}
da(pi_, pj_, s_, pis_) = 
  Integrate((mui*(1 - cdf(Max(pi, vj - pj + pi - s))) + (1 - mui)*(1 -
          cdf(Max(0, vj - pj) + Max(pi, pis + s))))*pdf(vj), {vj, vlo,
     vlo + 1});
da(0.2, 0.3, 0.1, 0.25) /. {vlo -> 0.1, mui -> 0.4}
d1(pi_, pj_, s_, pis_) = 
  Integrate((1 - cdf1(Max(pi, vj - pj + pi - s)))*pdf1(vj), {vj, vlo, 
vlo + 1});
d1(0.2, 0.3, 0.1, 0.25) /. {vlo -> 0.1, mui -> 0.4}
d1a(pi_, pj_, s_, pis_) = 
  Integrate((mui*(1 - cdf1(Max(pi, vj - pj + pi - s))) + (1 - 
         mui)*(1 - cdf1(Max(0, vj - pj) + Max(pi, pis + s))))*
    pdf1(vj), {vj, vlo, vlo + 1});
d1a(0.2, 0.3, 0.1, 0.25) /. {vlo -> 0.1, mui -> 0.4}
d2(pi_, pj_, s_, pis_) = 
  Integrate((1 - cdf2(Max(pi, vj - pj + pi - s)))*pdf2(vj), {vj, vlo, 
    vlo + 1});
d2(0.2, 0.3, 0.1, 0.25) /. {vlo -> 0.1, mui -> 0.4}
d2a(pi_, pj_, s_, pis_) = 
  Integrate((mui*(1 - cdf2(Max(pi, vj - pj + pi - s))) + (1 - 
         mui)*(1 - cdf2(Max(0, vj - pj) + Max(pi, pis + s))))*
   pdf2(vj), {vj, vlo, vlo + 1});
   d2a(0.2, 0.3, 0.1, 0.25) /. {vlo -> 0.1, mui -> 0.4}

The use of Simplify`PWToUnitStep @ does not seem to accelerate integration.

integration: an integral part involved in the calculation of the laws of power that arise from the Abelian sand pile

Dice $ E $ a finite subset of $ mathbb {Z} $ Y $ varphi: E to mathbb {N} ^ * $, we can define a graph in which the laws of power that emerge from the Abelian sand pile in this graph are related to the asymptotic behavior of

$ f (z) = displaystyle int_ {0} ^ {2 pi} frac { mathrm {d} x} {1-zSg (x)} $ when $ z a 1 –,

where $ g: x in (0, 2 pi) mapsto displaystyle sum _ { delta in E} varphi ( delta) e ^ {i x delta} $ Y $ displaystyle S = sum _ { delta in E} varphi ( delta) $.

Yes $ displaystyle sum _ { delta in E} varphi ( delta) delta = 0 $ then $ f (z) sim_ {z a 1} frac {C} { sqrt {1-z}} $. I got it by naively replacing $ exp $ with $ 1 + x + x ^ 2 $ but I don't have the analytical tools necessary to determine the equivalence in the general case and justify the use of the Taylor series in the integral.

How to formally determine the asymptotic behavior of $ f (z) $ when $ z mapsto 1 ^ – $ ?

calculation: identification of an unusual curve (parametric)

I am looking for the parametric equations for the curve shown: I know $ and (t) $it is $ y (t) = frac {1-φ ^ {t}} {1-φ} $; here $ φ it is $ frac {-1+ sqrt5} {2} $. So all I need is to find $ x (t) $, which will also be defined in terms of $ φ.

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Thank you all!

calculation and analysis – Teach Mathematica the analytical continuation of the gamma function

If I ask Mathica to calculate the gamma function for me

Integrate(Exp(-s) s^(a - 1), {s, 0, Infinity})

Obediently come back to me

ConditionalExpression(Gamma(a), Re(a) > 0)

However, we all know that we can continue analytically this function in the left semiplane (safe for non-positive integers). Therefore, I would like Mathica to understand that when I write

Integrate(Exp(-s) s^(-3/2), {s, 0, Infinity})

I want him to come back $ Gamma (-1/2) = -2 sqrt { pi} $ to me. (Although I understand that mathematically speaking it may not be strictly speaking correct since the integral is not really convergent). Similarly, I want you to understand that $ int_0 ^ infty e ^ {- s z} s ^ a-1} ds ; ; “ = "; ; z ^ – a} ; Gamma (a) $ when $ to $ It is in the middle left plane.

How do I efficiently teach Mathica to recognize this analytical continuation?

Thanks for your help!

terminology: trace versus calculation vs. execution

Many people use the terms "trace", "execution", "calculation", "run", "walk" … interchangeably when they talk about state machines with labeled transitions. Does any work distinguish between these concepts? For example, whose work says that XXX is a sequence of states, YYY is a sequence of labels and ZZZ is a sequence of states and labels between them, for different pairs XXX, YYY, ZZZ in {trace, execution, calculation, run, walk , …}?

Here, a state machine is simply a directed graphic rooted (where many roots are arbitrarily allowed, which are interpreted as initial states) with borders labeled with letters of a fixed alphabet.