## Separation of variables in vector components

I have a function f in two variables y and theta, where y = z ^ 2 + x ^ 2. There are numerical values ​​for x, z and theta (x and z are the dimensions). I have to do calculations to get the speed. Therefore the speed vector has to be expressed separately in two components x and z. The following is a simple example of my task:

``````f(y_,θ_) = a y Cos θ + b y^2 (1 - 3 Cos θ)

v(y_, θ_) = f(y, θ)/gradf(y, θ)
``````

I have taken the following path, but it is not correct.

``````v((z^2 + x^2) _, θ_) = f(y, θ)/gradf(y, θ)
``````

I need the separation of variables as components of the vector.

Thank you very much.

## linear algebra – be \$ C[0,1]Let \$ be the real vector space of all continuous real-valued functions

Given that Let $$C (0.1)$$ be the real vector space of all continuous real-valued functions $$(0.1)$$ To let $$T$$ be the linear operator on it and defined by

$$(Tf) (x) = int_0 ^ 1 sin (x + y) f (y) ; dy ; ; ; x in (0.1)$$

you will find the dimensions of the area space from $$T$$

I tried it-The specified transformation is an operator, so it is defined as follows $$T: C (0.1) rightarrow C (0.1)$$ After that, I have no idea how to solve it. Please give me a hint

Thank you very much

## Vector valued functions and tangents [closed]

Need help with these 3 questions. Can't seem to find how to change the curve equation to a vector value.

A particle moves along the curve 2y ^ 3 = 3x ^ 2 at a constant speed of 10 units / second.

a) Find the slope of the tangent at the point (4 / sqrt (3), 2)

b) Use the slope of the tangent found in a) to determine the unit direction vector at the point (4 / sqrt (3), 2).

c) At a speed of 10 units / second, enter the speed vector of this particle here.

Any help is appreciated, thanks!

## List manipulation – fixed point of a vector

It seems to me that FixedPoint is designed to work with a certain value, but what if you want it to work with a vector instead?

``````f[vec_]:=Exp[-vec]/Total[Exp[-vec]]
``````

I want to find a vector of the probabilities vec = {p [1], p [2], …, p [n]} so that:

``````vec==f[mat.vec],
``````

where each p [i]> 0 and the sum of p [i] 1.

Is there a way to do this in general? Or let's take a certain matrix:

``````test={{0.5, 0.44, 0.58}, {0.56, 0.5, 0.41}, {0.42, 0.59, 0.5}}
``````

Can I find a vector of probabilities {p1, p2, p3} that works here?

This appears to be a FixedPoint problem, but I would settle for any solution, e.g. B. with NSolve or a module / block. I puzzled over it for a while, so any help would be appreciated.

## c ++ – delete many vector elements when running with & # 39; auto & # 39;

Let's say I did `vector` of pairs where each `pair` Corresponds to the indices (row and column) of a particular matrix that I am working on

``````vector> vec;
``````

I wanted to go with `auto`, go through the entire vector and immediately delete all pairs that meet certain conditions, such as something like

``````for (auto& x : vec) {
if (x.first == x.second) {
vec.erase(x);
}
}
``````

but it doesn't work as i suppose `vec.erase()` should have an iterator as an argument and `x` is actually a `pair` that's an element of the vector `vec`not iterator. I've tried changing it in a few ways, but I'm not sure how to deal with container elements `auto` works exactly and how can I fix it.

Can I just change the code above to make it work and delete multiple vector elements as I work through it? `auto`? Or should I change my approach?

At the moment it is just a vector of pairs, but it will be much worse later, so I would love to use it `auto` for the sake of simplicity.

## Draw a vector field from spherical coordinates

I'm trying to draw an integral in spherical coordinates, but I'm a bit lost. I think my only problem is switching to Cartesian. Everything I saw about it went a bit over the head. Here is my code (sorry for formatting):

$$vec {E} = sigma k int_0 ^ R int_0 ^ {2 pi} ( frac {(r sin theta cos phi-r & # 39; cos phi & # 39;) textbf {i} + (r sin theta sin phi-r & # 39; sin phi & # 39;) textbf {j} + (r cos theta) textbf {k}} {( r ^ 2 + r & # 39; ^ 2-2rr & # 39; cos phi cos phi & # 39; sin theta-2rr & # 39; sin theta sin phi sin phi & # 39;) ^ {3/2}}) r & # 39; d phi & # 39; dr & # 39;$$

``````(ScriptR)((Phi)_, r_, (Theta)_, (Phi)1_,
r1_) := {r*Sin((Theta))*Cos((Phi)) - r1*Cos((Phi)1),
r*Sin((Theta))*Sin((Phi)) - r1*Sin((Phi)1), r*Cos((Theta))};

(ScriptR)Norm((Phi)_, r_, (Theta)_, (Phi)1_, r1_) :=
Sqrt((r*Sin((Theta))*Cos((Phi)) -
r1*Cos((Phi)1))^2 + (r*Sin((Theta))*Sin((Phi)) -
r1*Sin((Phi)1))^2 + (r*Cos((Theta)))^2) // Simplify;

ele((Phi)_?NumericQ, r_?NumericQ, (Theta)_?NumericQ) := (Sigma)*k*
NIntegrate(((ScriptR)((Phi), r, (Theta), (Phi)1,
r1)/(ScriptR)Norm((Phi), r, (Theta), (Phi)1, r1)^3)*
r1, {(Phi)1, 0, 2*(Pi)}, {r1, 0, R});

VectorPlot3D(
ele((Phi), r, (Theta)) /. {(Sigma) -> 200, k -> 9*10^9, R -> 0.5,
r*Sin((Theta))*Cos((Phi)) -> x, r*Sin((Theta))*Sin((Phi)) -> y,
r*Cos((Theta)) -> z}, {x, -1, 1}, {y, -1, 1}, {z, -1, 1})
``````

I'm concerned that part of my problem is in the function I created, but I'm very sure that calling "Replace All" in this way is not a viable method for transforming coordinates.

Edit: Added integral that I am trying to solve. The (ScriptR) is the vector in the numerator, and the (ScriptR) norm is the cube root of the denominator

## c ++ – Find the missing element in the vector

I solve this simple challenge:

For a vector A with N unique elements larger than 0 and smaller than N + 1, find the missing element. Example:

A = (1,3,2,5) -> missing number 4
A = (1,3,5,4) -> missing number 2

I came to the following solution. I am interested in thoughts and ideas on how to write it as expressively as possible:

Option 1, compact, but not very meaningful:

``````int solution_1(std::vector &v) {
sort(v.begin(), v.end());
for (std::size_t i = 0; i < v.size(); ++i) {
if (v(i) != i+1) return i+1;
}
return v.size()+1;
}

``````

Option 2

``````int solution_2(std::vector &v) {
sort(v.begin(), v.end());
auto missing_element = std::find_if(
v.begin(), v.end(),
(index=1)(auto& element) mutable {
if (element != index++) {
return true;
} else {
return false;
}
});
if (missing_element == v.end()) {
return v.size() + 1;
} else {
return *missing_element - 1;
}
}
``````

Any ideas on how to improve this or make it more expressive?

## Can you help formulate a problem with vector projections?

I have some vectors and I am trying to get a certain combination of them. I managed to work out the solution using trigonometry, but it would be cleaner if I could only do this with vector arithmetic and vector projections.

I have three vectors: $$q, n$$ and $$v$$, I want to get the projection from $$q$$ to the hyperplane (which runs through the origin) to which $$n$$ is normal "along" the direction $$v$$, In other words, I want a vector $$u$$ in order to:

• $$u cdot n = 0$$ ($$u$$ lies in the hyperplane)
• $$q-u = lambda v$$ ($$u$$ lies "along" the direction of $$v$$ of $$q$$)

Here is an illustration. Note that $$q$$ and $$u$$ are to scale, but $$v$$ and $$n$$ are of any (e.g. unit) length. In other words, $$q + v neq u$$ generally. Also note that $$q$$. $$n$$ and $$v$$ can be of a high dimension.

If it is helpful, use trigonometry and the projection of $$q$$ on to $$n$$, I got that:

$$u = q – frac {q cdot n} {v cdot n} v$$

But I only want to use vector operations in my arguments to get this result if possible. How can we use geometric arguments to get this result from projections and vector arithmetic?

## Draw – Can I draw a time-dependent 3D vector?

I wanted to know if I could draw a 3D vector (with x, y and z components) in which each component is time dependent. For more context, I would like to draw the function:

E (t) = 3cos (ωt) (x_hat) + (3cos (ωt) – 4sin (ωt)) (y_hat) – 6cos (ωt – π / 4) (z_hat)

Which is represented by a pointer in the time domain. I think the best way is to keep the time constant and plot the function at a specific point in time, but I've also seen my professor create a chart that gradually plots a variable n by itself or by moving one Bar changes. How do I do this or is there a better way to draw this function? Again, this is not part of my homework, but I want to create a diagram to better understand what I am doing for my homework and how it looks plotted.