Why is the gradient of my scalar potential not the same as its conservative vector field?

I'm fairly new to this so I must've made some obvious mistake.

I assumed:

P = [x, y]

F = [f1(P), f2(P)]

... where F is a conservative vector field. Next, I tried to set up two different parameter equations for the same smooth curve over which I wanted to do a line integral of the vector field.

P*(t) = [t, (y/x) t ]

P**(u) = [ (x/y) u, u]

That's how my line integral became:

phi(P) = ∫ (0 to x) [ f1[P*(t)] + (y/x) f2[P*(t)] ] dt

= ∫ (0 to y) [ (x/y) f1[P**(u)] + f2[P**(u)] ] du

Which yields partial derivatives:

Dx phi(P) = f1(P) + y/x f2(P)

Dy phi(P) = x/y f1(P) + f2(P)

It seems I'm some strange terms shy of what I wanted, could somebody explain what this means or point me to what I might be doing wrong?

Note:

This is not a homework question, nor from an examination. I'm an independent student of computer science in Belgium, following my local university's curriculum and using their courses. I've uploaded a relevant page from my Analysis III course [ https://imgur.com/Htc6dnm ] (the equivalent of American Calculus III with some proof-based analysis). My particular question to you, however, concerns a problem of my own fabrication that I stumbled upon while playing around with some of the concepts in the course -- that is to say, it's not explicitly mentioned anywhere in the course.

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Get ready everyone (Desmos graph credit to u/scalar-field)
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Recover scalar field/image from its X Y gradients

Hi,

I have a single channel image from which I can compute its vertical and horizontal gradients. I would like to make some operations in the gradient domain and subsequently recover back the scalar field (image) which results after the gradient modification. Any idea how to do this? I know if I integrate the modified gradient I can get back the function up to a constant but I would have two different constants C_x and C_y from the partial X and Y derivatives. Also, I don't have an intuition of how to "integrate" a discrete vector field as the gradient.

Thanks!

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Animation of Line Integrals of Scalar Fields with 3b1b's manim. I've also made a video on it (Link in the comments), would be glad to receive any feedback. v.redd.it/uawlrhek8q961
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Animation of Line Integrals of Scalar Fields. I've also made a video on it (Link in the comments), would be glad to receive any feedback. v.redd.it/xn79674poq961
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Let’s say I’ve 9 complex scalar fields governed by coupled nonlinear PDEs. And Hamiltonian density contains contributions from all of these 9 fields. How do I get the dispersion relation(s) for such a system?

Edit: forgot to mention that these scalar fields don’t evolve unitarily i.e. their norm isn’t a conserved charge of the system. So I think i*dphi/dt is not equal to omega * phi

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I have an assignment to do some derivations and take some limits of compton scattering in scalar qed. Specifically the use of Mendelstam variables is required. This is more to verify my derivation than to copy. Any help appreciated.

Edit: found the solution in Quantum Field Theory and the Standard Model by Mathew D. Schwartz. The process is actually pion scattering.

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I made an animation on Line Integrals of Scalar Fields & also made a video (Link in comments) sharing my perspective regarding the transition from integrating with "ds" to integrating with "dt" , which is essentially just rescaling the riemann sum. I'd be glad to hear your thoughts on this, thanks! v.redd.it/k15pbfkjsq961
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Animation of Line Integrals of Scalar Fields made with manim. I've also made a video on it (Link in the comments), would be glad to receive any feedback. v.redd.it/t72b8p8ugq961
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Animation of Line Integrals of Scalar Fields made with manim. I've also made a video on it (Link in the comments), would be glad to receive any feedback. v.redd.it/5qt636o1nq961
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Line Integrals of 2D Scalar Fields, made with 3b1b's shaders youtu.be/G28aYRu9aZI
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Animation of Line Integrals of Scalar Fields made with manim. I've also made a video on it (Link in the comments), would be glad to receive any feedback. v.redd.it/w107bdrtdq961
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Hello Mayans, sorry for a total newb question .... I'd like to spice an animated section up with a display of interpolated color-coded temperature values on the section plane. A little more precisely, my data are a surface object, and the temperature values on some discretely spaced point cloud that fills the interior of the object in a not very regular fashion. Now, instead of just slicing the object and displaying the section contour as usual, I'd like Maya to generate a mesh of the slicing contour on the slicing plane, and interpolate the temperature values on the vertices of that mesh from the closest points with known temperature, or maybe in a more advanced fashion using some inverse distance weighting or whatever. So far, I have imported the object and can animate the section, but I have no clue how to import the point cloud, how to import the temperature values on these points , how to make Maya mesh the slicing contour and interpolate the temperatures and color the slices meshes accordingly. Hope this is a clear description.

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Scalar and Vector Fields youtube.com/watch?v=ybNSE…
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Hi I’m trying to define a passive scalar location for my tank (which has no inlet) as a spherical ball (equation x^2 + y^2 + z^2 = 0.1). Essentially a sphere of radius sqrt(0.1) at origin. What would be the syntax for this?

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Is the line integral of a scalar field a vector?

Hi, I'm studying line integrals of a scalar field and I find two definitions:

https://preview.redd.it/4247aqb80rb51.png?width=570&format=png&auto=webp&s=df742d9382f6f1ed6bac17757be67098a193b6e6

and

https://preview.redd.it/ela59ui5zqb51.png?width=338&format=png&auto=webp&s=4c2832d637878c38deb875d316b789b7a723e5b4

The first is a vector and the second is a scalar. I'm confused. Are they both correct? Why are they different?

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Scalar function too slow - Calculated field in column based on another columns

I have a big query, (really big, 800+columns - can't really make it smaller) that I'm trying to build in the most efficient way.

I have some fields like this:

SELECT distinct(td.EVENT_ID), 
       td.MENU_HINT, 
       td.EVENT_NAME, 
       td.EVENT_DT AS Expr1003, 
       COUNT(td.EVENT_NAME) AS [Total Runners], 
       -- AS [WinnerPrice], 
       -- COUNT(td.ID) AS [WinnerCount],
       SUM(CASE WHEN td.BSP >= 1 AND td.BSP < 2 THEN td.BSP END) AS Sum1to2, 
       count(CASE WHEN td.BSP >= 1 AND td.BSP < 2 THEN td.BSP END) AS count1to2, 
       avg(CASE WHEN td.BSP >= 1 AND td.BSP < 2 THEN td.BSP END) AS average1to2,
       dbo.FN_SumWinnerRunners(td.EVENT_ID,34,55,0,7,12) AS [TEST FUNCTION],
       SUM(CASE WHEN (td.BSP >= 2 AND td.BSP < 3) AND td.WIN_LOSE =1 THEN td.BSP END) AS SumWinnerPrice2to3, 
       count(CASE WHEN (td.BSP >= 2 AND td.BSP < 3) AND td.WIN_LOSE =1 THEN td.BSP END) AS CountWinnerPrice2to3
       FROM tblData td 
GROUP BY td.EVENT_ID, 
         td.MENU_HINT, 
         td.EVENT_NAME, 
         td.EVENT_DT
ORDER BY [Total Runners] DESC;

And here a graphical one (click to enlarge):

https://preview.redd.it/5xpivq1wvum41.png?width=1039&format=png&auto=webp&s=c77b04efb616ad4c7909da16b8c39ace75ba429e

if you notice, I've tried, as I have several queries that are fairly similar, to use a function (dbo.FN_SumWinnerRunners(td.EVENT_ID,34,55,0,7,12)

the definition for it below:

CREATE FUNCTION [dbo].[FN_SumWinnerRunners](
                                @event_id    INT,   
                                @BSPFrom     SMALLINT, 
                                @BSPTo       SMALLINT, 
                                @Win_Lose    SMALLINT, 
                                @RunnersFrom SMALLINT, 
                                @RunnersTo   SMALLINT)
RETURNS FLOAT
AS
     BEGIN
         DECLARE @result FLOAT=
         (
             SELECT SUM(CASE
                            WHEN(td.BSP >= @BSPFrom
                                 AND td.BSP < @BSPTo)
                                AND td.WIN_LOSE = @Win_Lose
                            THEN td.BSP
                        END)
             FROM tblData td
             WHERE td.EVENT_ID = @event_id
             HAVING COUNT(td.EVENT_NAME) > @RunnersFrom
... keep reading on reddit ➑

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As this website and other textbooks shows, to pass from the microscopic to macroscopic version of Ohm's law, these assumptions are made:

β€’ E=βˆ†V/L which is only true if the field through the conductor is uniform.

β€’ J=I/A that due to its definition implies that the direction of the drift speed of electrons is the same as the normal vector of the conductor's surface.

But this can't be true considering we are assuming pJ=E and E, since the conductor isn't in electrostatic equilibrium anymore, is the field inside of the conductor generated by the current. So it makes sense for me that E can be considered uniform because is almost always tangencial to the path, but then again, the direction of J and E can't be the same to make these 2 assumption simultaneously, am I right? What am I missing in this analysis?

Also, I couldn't find anything mathematically rigorous on this particular issue, like how is it possible that we can have a vector and scalar form of Ohm's law that are equivalent? Is the law actually the module of J and E or why is the vector form which is correct?

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Happiness is a scalar field in the many dimensional phase space representing the state of the universe. Finding the point at which your happiness is maximum is very hard, taking infinite computation time in the general case. Instead, the best way to find happiness is to perform gradient descent. Looking at every small step you could take in the phase space from where you currently are, pick the one which changes your happiness the most. If you continue to do so, eventually you'll find a local maximum of your happiness field. This may not be the global maximum of the happiness function but at the very least it's a local maximum with the added benefit of being able to find it in finite time. Although if you're going to take this advice, remember time is also a dimension of the phase space and you are constantly taking steps through time so make sure to adjust for those.

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More of a physics question, but do you think there a typo in this question? I know a scalar field multiplied by a vector field returns a vector field, and the gradient of a vector field is a tensor, which we never covered in class. Am I missing something?
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