What is Normal: Definition and 1000 Discussions

In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point.
A normal vector may have length one (a unit vector) or its length may represent the curvature of the object (a curvature vector); its algebraic sign may indicate sides (interior or exterior).
In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles).
The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P is the set of vectors which are orthogonal to the tangent space at P.
Normal vectors are of special interest in the case of smooth curves and smooth surfaces.
The normal is often used in 3D computer graphics (notice the singular, as only one normal will be defined) to determine a surface's orientation toward a light source for flat shading, or the orientation of each of the surface's corners (vertices) to mimic a curved surface with Phong shading.

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  1. C

    Using Conjunctive Normal form to find when wff is true

    For this, Does someone please know how setting ##P_1## and ##P_2## true makes the CNF true? If I see ##P_2## true, then it ##(true + false)## since it is negated. Therefore, should they be setting ##P_1## true and ##P_2## false? Many thanks!
  2. gmax137

    I Ordering other than normal

    In another thread This has me curious about "ordering other than our normal ordering." What does this mean? I take it that "normal ordering (of integers)" is ... 0, 1, 2, 3... Do mathematicians consider alternate orderings like ...0, 2, 1, 3... That doesnt seem to make sense to me, that's...
  3. E

    Work done by a normal force (or rather, work NOT done)

    This question states that the normal force of the stairs on the woman does NO work. I do not understand how this can be. I would reason like this: The woman propels herself up the stairs using her legs. Her legs push down against the stairs, and the consequent normal force pushes upwards on her...
  4. dford

    Normal modes of four coupled oscillating masses (Kleppner and Kolenkow)

    This exercise comes from Kleppner and Kolenkow, 2nd ed., problem 6-3. I'm using a solution key as a study reference, but the solution key is coming to a pretty different conclusion. Mostly the issue is in the equations of motion for this system. I'm not sure if there's something I'm...
  5. baby_1

    The relation between the normal and the slope of a cylindrical curve

    As you can see in this picture: This explanation "relation between the normal and the slope of a curve" is formulated here: $$\frac{1}{\rho} \frac{d\rho }{d\psi }=\tan\left(\frac{\theta+\psi}{2}\right)$$ I got confused because I don't have the curve equation(regarding the slope of the curve...
  6. E

    Engineering Normal and Shearing Stress - Combined Loading

    Here is my combined loading: The book solution for normal and shear stresses respectively are: a) ##20.4~\text{MPa}, 14.34 ~\text{MPa} ## - I find both b) ##-21.5~\text{MPa}, \boxed{19.98~\text{MPa}}## - I find the normal stress, but I'm not getting the book answer for the shear stress...
  7. Remle

    Acceleration & Normal Force of a Box: 4 m/s2 & 418 N

    What is the acceleration of the box? Paper says the answer is 4 m/s2. What is the Normal force acting on the box? Paper says the answer is 418 N. I know that for most cases FN=Fg=W. So, by definition the "original" Normal force is 245.25 N (am I correct?) I calculated the Fay which is...
  8. Quasar100

    Normal Forces on a Sphere in a Non-vertical Groove

    neglect friction and motion (sliding) and G(sphere)=20N. In this question I reached two different result with two different solving method.But one of them is false according to answer key. My question is why first solving way is false? Because the first solution way makes sense to me. If we...
  9. S

    Thickness of normal shock zone (e.g. in nozzle)

    Under certain conditions, a supersonic flow in a nozzle will result in a "normal shock", an abrupt change in pressure and velocity. In the videos I've looked at, they draw the P and v graphs with a vertical step at that point. But in practice, I assume there will be a non-zero transition zone...
  10. StanislavD

    A Superconducting and normal electrons are not interchangeable

    An interesting paper in NATURE "A superconductor free of quasiparticles for seconds" https://www.nature.com/articles/s41567-021-01433-7 showing that superconducting (paired) electrons don't hop into normal states for seconds. The measurement device detects single pair-breaking-events for a large...
  11. S

    The FBD Mystery: Why Isn't Normal Force Along Radial Line?

    so this is what the FBD is.... but to be fair, to me this one looks as if the normal force in the direction of the radial line, yet it isn't???? here in the solution, it's not along the radial line, whys that???
  12. S

    Angle between normal force and radial line for cylindrical coordinates

    so I was wondering. there is this normal force on the can from the path. And there's this formula to find the angle between the radial line and the tangent or also between the normal force and either the radial or theta axis. the formula is ##\psi = r/dr/d\theta##. The thing is that here they...
  13. Eclair_de_XII

    Is it normal for a student to get frustrated learning a coding language?

    ... after many failed attempts at figuring it out? Is it normal for him to soon after declare himself done trying to figure out the solution and the language, only to return to it after only a short period of distancing himself from it by a short walk or some number of hours goofing off? Is it...
  14. Onyx

    B Calculate Unit Normal Vector for Metric Tensor

    How do I calculate the unit normal vector for any metric tensor?
  15. arhg

    A Normal force and force of friction inside a tube

    Dear all, Me and some colleagues (non-physicists) are discussing how force works when passing a cylinder (which we are holding) into a narrow tube. As we insert more of the cylinder into the tube, the force we are exerting is increasing. My theory is that the normal force is increasing and his...
  16. Astronuc

    English is not normal, says John McWhorter

    I just stumbled across an essay by linguist John McWhorter from 2015. https://aeon.co/essays/why-is-english-so-weirdly-different-from-other-languages
  17. Crunge

    The maximal normal force (Friction with slanted surface)

    The problem that I immediately ran into was how I would calculate N without knowing Fmax. I didn't think the y-component of N would simply be the same magnitude as mg. After being stuck for a good while I even tested if it was, by dividing the magnitude of mg with cosθ, which of course ended up...
  18. Idontknowhatimdoing

    Normal force at the top of a vertical loop -- Circular Motion Dynamics

    From the equation for centripetal force, I can see that the centripetal force is proportional to v^2. Does this have something to do with why there is a normal force at the top? Does the velocity of the object require there to be a normal force? If so, why is that the case?
  19. A

    A Discrete type normal distribution

    The following is given: $$\displaystyle P(K = k) = \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(k -\mu \right)^{2}}{\sigma ^{2}}}}{\sigma ~\sqrt{\pi }}$$ How can you prove that the following equalities are correct? $$\displaystyle \sum _{k=-\infty }^{\infty }1/2\,{\frac {...
  20. N

    [Mathematical logic] prenex normal form and skolem normal form

    The attached picture below is the note I solved halfway through. Please tell me the entire process of getting to the correct answer.
  21. F

    I Reaction force of and its relation to normal force and friction

    Hello, When we consider a block sitting on a surface, the gravitational force ##W## and the normal force ##F_N## are applied to the block. Both equal i magnitude and opposite in direction. We call the normal force the reaction force exerted by the surface on the block. Now we consider the...
  22. A

    A A different discrete normal distribution

    In the article A Discrete Normal Distribution of Dilip Roy in the journal COMMUNICATION IN STATISTICS Theory and methods Vol. 32, no. 10, pp. 1871-1883, 2003 one can read: A discrete normal (##dNormal##) variate, ##dX##, can be viewed as the discrete concentration of the normal variate ##X##...
  23. H

    Improper integral of a normal function

    I'm trying to solve an improper integral, but I'm not familiar with this kind of integral. ##\int_{-\infty}^{\infty} (xa^3 e^{-x^2} + ab e^{-x^2}) dx## a and b are both constants. From what I found ##\int_{-\infty}^{\infty} d e^{-u^2} dx = \sqrt{\pi}##, where d is a constant and...
  24. E

    B Normal Force Discrepancy for Wedge Vs. Ramp

    I have seen a few posts on this subject before, but none have really answered my question. For clarity, I will refer to the 1st example as a wedge, and the second as a ramp (although both are of course inclined planes). With both examples that I outline below, we will assume no friction, and a...
  25. N

    I Bell's circuit beginner's question: Sampling alters, - what's normal?

    Yes, I should hit the books more, so forgive the basic question. I take it normality is known by observing unsplit photon spin ? But how can one then exclude that split photons in themselves might have different probability outcomes ? Thx much in advance. (Please pardon that only...
  26. S

    Normal vector of plane

    I know the normal of plane ABQP is -2i - j + 3k but I don't know how to prove that 2i + j + 3k is the normal vector of plane CDPQ Thanks
  27. V

    How can the force of a normal reaction be electromagnetic?

    I do understand that gravitational the electromagnetic force between two electrons or protons is very large compared to the gravitational force between them. I can see this by looking at the equation of gravitational force (##F= \frac {Gm_1m_2} {r^2}##) and the equation of electrical force given...
  28. G

    Tong QFT sheet 2, question 6: Normal ordering of the angular momentum operator

    My attempt/questions: I use ##T^{0i} = \dot{\phi}\partial^i \phi##, ##\dot{\phi} = \pi##, and antisymmetry of ##Q_i## to get: ##Q_i = 2\epsilon_{ijk}\int d^3x [x^j \partial^k \phi(\vec{x})] \pi(\vec{x})##. I then plug in the expansions for ##\phi(\vec{x})## and ##\pi(\vec{x})## and multiply...
  29. Rikudo

    Distribution of normal force in this problem

    It is crystal clear that we need torque equation to solve this. But, in order to do so, I need to know where the normal force is located. As far as I'm concerned, normal force is not distributed equally. If this is true, then I suppose this problem is unsolvable? (Though the book says thay it is...
  30. hilbert2

    I Error tolerant normal mode frequency

    If a Hookean spring-mass system is made from one mass and a spring, to produce a system with a particular oscillation frequency, it's not a problem to use the propagation of errors concept to find how this frequency responds to small errors in the mass and spring constant. If a chain of...
  31. T

    Knowing when to decompose weight vector vs. normal vector

    Good afternoon everyone, I have a question on Newton's 2nd Law regarding objects on a generic incline. Take for example, a car on a banked curve: Here in the picture I've provided, you can see that the normal force has been decomposed into the x and y components via sine and cosine of the...
  32. G

    I How Can Outward Normal Vector Point Inwards?

    I'm following 《A First Course In General Relativity》.On page 72,it says"If the surface is spacelike,the outward normal vector points outwards.If the surface is timelike,however,the outward normal vector points inwards"I wonder why and how?
  33. chwala

    Solve the problem that involves Normal distribution

    My interest is on part (c), My take, ##Z=\dfrac{160−200}{60}=−0.666666## ##Pr(−0.66666)=0.3546## ##⇒\dfrac{x_1-200}{60}=1.05## ##x_1=63+200=263## Yes, i am aware that they want the answer to ##5## significant figures...i just wanted to check the alternative method... Appreciate your insight...
  34. A

    A A discrete version of the normal distribution

    I have the following function for the normal distribution: $$\displaystyle f \left(x \right) = \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -\mu \right)^{2}}{\sigma ^{2}}}}{\sigma ~\sqrt{\pi }}$$ How can the following integrals be equal to their sums? $$\displaystyle \int_{-\infty...
  35. S

    Normal vector of an embedding surface

    I will only care about the ##t## and ##x## coordinates so that ##(t, z, x, x_i) \rightarrow (t,x)##. The normal vector is given by, ##n^\mu = g^{\mu\nu} \partial_\nu S ## How do I calculate ##n^\mu## in terms of ##U## given that the surface is written in terms of ##t## and ##x##? Also, after...
  36. J

    Normal modes of oscillation

    The first part is trivial not sure where to go on the second part.
  37. LCSphysicist

    Normal modes: Spring and pendulum

    I was doing the exercise as follows: I am not sure if you agree with me, but i disagree with the solution given. I was expecting that the kinect energy of the mass ##m## (##T_2##) should be $$T_2 = \frac{m((\dot q+lcos(\theta)\dot \theta)^2 + (lsin(\theta) \dot \theta)^2)}{2}$$ I could be...
  38. B

    I Does Axler's Spectral Theorem Imply Normal Matrices?

    Going through Axler's awful book on linear algebra. The complex spectral theorem (for operator T on vector space V) states that the following are equivalent: 1) T is normal 2) V has an orthonormal basis consisting of eigenvectors of T and 3) the matrix representation of T is diagonal with...
  39. A

    A The normal equivalent for a discrete random variable

    De normal distribution has the following form: $$\displaystyle f \left(x \right) \, = \,\frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -\nu \right)^{2}}{\tau ^{2}}}}{\tau ~\sqrt{\pi }}$$ and it's integral is equal to one: $$\displaystyle \int_{-\infty }^{\infty }\!1/2\,{\frac {...
  40. rudransh verma

    Coefficient of friction and normal force

    The answer should be no change but we know ##F=ma##. In this eqn when acceleration increases mass decreases for same force. So why not here? If normal is doubled ##\mu## should be halved.
  41. Addez123

    Find all points where surface normal is perpendicular to plane

    a. I solved a but I don't fully understand how it works. $$z = f_x'(1, -1)(x -1) + f_y'(1, -1)(y+1) = 2(x-1) + 3(y+1)$$ Eitherway it's b that's my issue. I can find the gradient of both plane and surface, but trying to do "dot-product of both normals = 1" will give an equation involving two...
  42. A

    I Normal force and details of the Earth holding up an object

    It's a simple application of Newton's third law to show that the Earth indeed does accelerate towards an object as it falls towards earth. M_o is the mass of the object M_e is the mass of the earth From the third law (and ignoring air drag): M_e * a_e - M_o*g = 0 (with a up-positive...
  43. Moara

    Engineering Maximum normal stress and shear stress

    First, I am trying to find the external reactions in A and B, but I have only one equation relating ##V_A## and ##V_B##, what other relation could I use ? Once I find the reactions, I can find the external moment as well. Then, I may draw the diagram of moments in each cross section and then...
  44. WMDhamnekar

    MHB How to prove the normal distribution tail inequality for large x ?

    What is the meaning of this proof? What is the meaning of last statement of this proof? How to prove lemma (7.1)? or How to answer problem 1 given below?
  45. Tapias5000

    How to calculate the normal stress of a Howe reinforcement?

    This is the structure I already made the calculation of all the bars T = tension and C = compression, these are the results. now I am asked to calculate the normal stress in all the bars but I don't understand where to start, could you tell me how? here is the diagram of the first node but I...
  46. Mr_Allod

    Analysing the Normal Modes and Dynamics of a Cluster of Atoms

    I am trying to analyse the dynamics of a cluster of 79 atoms. The system can be described with: ##\omega^2 \vec x = \tilde D\vec x## Where ##\omega^2## (the eigenvalues) are the squares of the vibration frequencies for each mode of motion, ##\tilde D## is the "dynamical matrix" which is a...
  47. karush

    MHB  standard normal sscurve P(Z le t) =0.9

    b 90\% of the insects die after t hours. (i) Represent this information on a standard normal curve diagram, indicating clearly the area representing 90\% (ii) Find the value of \textbf{t}. $P(Z\le t) =0.9\quad Z = 1.282\quad t=57+(4.4(1.282))=62.64$ hours \begin{tikzpicture}[scale=0.6]...
  48. L

    Transforming to a Normal Form (PDE)

    I don't know how to solve for u(x,y) from where I left of after 5.
  49. J

    A Why we can perform normal ordering?

    As explained in the summary, it seems that the commutators of some operators (creation and anihilation) can be ignored when quantising the hamiltonian of the Klein Gordon Field. I wonder why we are allowed to do such a thing. Is that possible because we are solely within a semiquantum...