What is Geometric interpretation: Definition and 38 Discussions

Geometric art is a phase of Greek art, characterized largely by geometric motifs in vase painting, that flourished towards the end of the Greek Dark Ages, c. 900–700 BC. Its center was in Athens, and from there the style spread among the trading cities of the Aegean. The Greek Dark Ages lasted from c. 1100 to 750 BC and include two periods, the Protogeometric period and the Geometric period (or Geometric art), in reference to the characteristic pottery style. The vases had various uses or purposes within Greek society, including, but not limited to, funerary vases and symposium vases.

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  1. G

    I Geometric Interpretation of Turbulence

    I would like to give a geometric interpretation to turbulence. Let's take into consideration for example a Poiseuille flow. The velocity profile resembles a parabolic bullet. As the particles are pushed by other layers of particles, then it must be that in addition to their translation, they...
  2. sergiokapone

    A Quantum superposition: Is a problem of space-time worldview?

    One of the paradoxical principles in Quantum Physics is the principle of quantum superposition, since in quantum theory we are not really talking about the superposition of waves or oscillations, but about the superposition of states. A classic example demonstrating the phenomenon of quantum...
  3. K

    What is the geometric approach to mathematical research?

    I read this article History of James Clerk Maxwell and it talks about Maxwell and Dirac also at some point. It is said that Maxwell thought geometrically, and also Dirac said he thought of de Sitter Space geometrically. They say their approach to mathematics is geometric. I see this mentioned...
  4. V

    Second reflection angle of incidences in 3D

    Given Theta1(angle of incidence) and alpha1(azimuth angle). how do we obtain the second reflection angle theta3 and alpha3? Assuming the surface to be a mirror reflection(theta1 = theta2). Need an equation when varied the incident angles we would obtain the second reflection angles or a method...
  5. M

    MHB Which is the geometric interpretation?

    Hey! :o Which is the geometric interpretation of the following maps? $$v\mapsto \begin{pmatrix} 0&-1&0\\ 1&0&0\\ 0&0&-1\end{pmatrix}v$$ and $$v\mapsto \begin{pmatrix} 1& 0&0\\0&\frac{1}{2} &-\frac{\sqrt{3}}{2}\\ 0&\frac{\sqrt{3}}{2}&\frac{1}{2}\end{pmatrix}v$$
  6. M

    MHB Geometric interpretation of maps

    Hey! :o We have the below maps: $f_1:\mathbb{R}^2\rightarrow \mathbb{R}^2, \ \ \begin{pmatrix}x \\ y\end{pmatrix}\mapsto \begin{pmatrix}-x \\ -y\end{pmatrix}$ $f_2:\mathbb{R}^3\rightarrow \mathbb{R}^3, \ \ \begin{pmatrix}x \\ y\\ z\end{pmatrix}\mapsto \begin{pmatrix}x \\ -y\\...
  7. P

    I Grover algorithm geometric interpretation

    Good day everybody, I'm currently working on the Grover algorithm. You can also illustrate this process geometrically and that's exactly what I have a question for. In my literary literature one obtains a uniform superposition by applying the Hadamard transformation to N-qubits. So far that's...
  8. Math Amateur

    MHB Geometric Interpretation of k-Forms from H&H's Vector Calculus

    I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard. I am currently focused on Chapter 6: Forms and Vector Calculus ... I need some help in order to understand some notes by H&H following Figure 6.1.6 ...
  9. C

    I Is there a geometric interpretation of orthogonal functions?

    Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...
  10. W

    Geometric interpretation of complex equation

    Homework Statement $$z^2 + z|z| + |z|^2=0$$ The locus of ##z## represents- a) Circle b) Ellipse c) Pair of Straight Lines d) None of these Homework Equations ##z\bar{z} = |z|^2## The Attempt at a Solution Let ##z = r(cosx + isinx)## Using this in the given equation ##r^2(cos2x + isin2x) +...
  11. M

    MHB What is a geometric interpretation of all these information?

    Hey! :o We have the tableau $\begin{pmatrix} \left.\begin{matrix} 1 & 0 & \alpha \\ 0 & 1 & \beta \\ 0 & 0 & 0 \end{matrix}\right|\begin{matrix} c\\ d\\ 0 \end{matrix} \end{pmatrix}$ Since there is a zero-row, we conclude that the column vectors are linearly dependent. The number...
  12. J

    I What's the geometric interpretation of the trace of a matrix

    Hello, I was just wondering if there is a geometric interpretation of the trace in the same way that the determinant is the volume of the vectors that make up a parallelepiped. Thanks!
  13. Markus Hanke

    I Geometric Interpretation of Einstein Tensor

    Is there a simple geometric interpretation of the Einstein tensor ? I know about its algebraic definitions ( i.e. via contraction of Riemann's double dual, as a combination of Ricci tensor and Ricci scalar etc etc ), but I am finding it hard to actually understand it geometrically...
  14. L

    Geometric Interpretation of complex numbesr

    z1,z2,z3 are distinct complex numbers, prove that they are the vertices of an equilateral triangle if and only if the following relation is satisfied: z1^2+z2^2+z3^2=z1.z2+z2.z3+z3.z1 so i shall show that |z1-z2|=|z1-z3|=|z2-z3|but i do not know how to start.
  15. Shackleford

    What is geometric interpretation of this equality?

    Homework Statement Prove that for any a, b ∈ ℂ, |a - b|2 + |a + b|2 = 2(|a|2 + |b|2). Homework Equations |a|2 = aa* (a - b)* = (a* - b*) (a + b)* = (a* + b*) * = complex conjugate The Attempt at a Solution I've already shown that the relation is true. I'm not quite sure what the...
  16. J

    Geometric interpretation for d²f/dxdy

    If the following integral: $$\\ \iint\limits_{a\;c}^{b\;d} f(x,y) dxdy$$ represents: So which is the geometric interpretation for ##f_{xy}(x_0, y_0)## ?
  17. N

    Geometric Interpretation of VSEPR Theory

    I am making a Geometric model for VESPR theory, which states that valence electron pairs are mutually repulsive, and therefore adopt a position which minimizes this, which is the position at which they are farthest apart, still in their orbitals. For example, the 2 electron pairs on either side...
  18. E

    What is the geometric interpretation of the vector triple product?

    The interpretation of the vector product is the area of the parallelogram with sides made up of a and b and the scalar triple product is the volume of the parallelpiped with sides a, b, and c, but what is the interpretation of the vector triple product. Is it just simply the area of the...
  19. MexChemE

    Geometric interpretation of partial derivatives

    Good afternoon guys! I have some doubts about partial derivatives. The other day, my analytic geometry professor told us that slopes do not exist in three-dimensional space. If that's the case, then what does a partial derivative represent? Given that the derivative of a function with respect to...
  20. B

    Non-Integrability of a Pfaffian - Geometric Interpretation?

    The question of Solving a Pfaffian ODE can be interpreted as the question of finding the family of surfaces U = c perpendicular to a surface f generated by the vector field $$F(x,y,z) = (P(x,y,z),Q(x,y,z),R(x,y,z))$$ At each point, the gradient of the family of surfaces U = c will either...
  21. shounakbhatta

    Orthogonal set - Geometric interpretation

    Orthogonal set -- Geometric interpretation Hello, If we have two vectors u,v then in an inner product space, they are said to be orthogonal if <u,v>=0. Well, orthogonal means perpendicular in Euclidean space, i.e. 90 degrees. How <u,v> becomes zero. Secondly, if I have three vectors...
  22. mnb96

    Geometric interpretation of metric tensor

    Hello, can anyone suggest a geometric interpretation of the metric tensor? I am also interested to know how we could "derive" the metric tensor (i.e. the matrix <ai,aj>) from some geometric considerations that we impose.
  23. A

    Geometric Interpretation of √(x^2+y^2+z^2) and its Derivative

    Let r = √(x^2+y^2+z^2) One can easily show that \nablar= \vec{r}/r. But I'm having a hard time understanding what this means geometrically - who can help? :)
  24. V

    Geometric interpretation of an equation

    Homework Statement x,y, z are vectors in R^n. We have the equation: ax +by +cz, where a,b,c are constants such that a+b+c=1, and a,b,c>=0 What is the geometric interpretation of the equation? Homework Equations sv + tu, where u,v are vectors in R^n and s,t are constants such that...
  25. B

    Geometric Interpretation of (lower) Cohomology?

    Hi, All: Just curious to know if there is an interpretation for lower cohomology that is as "nice", as that of the lower fundamental groups, i.e., Pi_0(X) =0 if X is path-connected (continuous maps from S^0:={-1,1} into a space X are constant), and Pi_1(X)=0 if X is...
  26. S

    Geometric interpretation of \int x f'(x)

    I was reading Tom Apostol's expostion of Euler's Summation Formula ( http://www.jstor.org/pss/2589145) and it occurred to me that it would be convenient to visualize \int_a^b x f'(x) geometrically. In that article, it arises from integration by parts: \int_a^b f(x) dx = |_a^b x f(x)...
  27. F

    Geometric interpretation of the spacetime invariant

    For a euclidean space, the interval between 2 events (one at the origin) is defined by the equation: L^2=x^2 + y^2 The graph of this equation is a circle for which all points on the circle are separated by the distance L from the origin. For space-time, the interval between 2 events is...
  28. icystrike

    Geometric interpretation and vector problem

    Homework Statement They were asking for the geometric interpretation and the says its triangular prism with infinite right angles. I don't understand what they mean by that. Homework Equations The Attempt at a Solution
  29. P

    Geometric interpretation of a given Alexandrov compactification

    What is the Alexandrov compactification of the following set and give the geometric interpretation of it: [(x,y): x^2-y^2>=1, x>0] that is, the right part of the hyperbola along with the point in it. This is a question from my todays exam in topology. I wrote that the given set is...
  30. D

    Help with geometric interpretation of 1-form

    I am currently reading the special relativity section in Goldstein's Classical, and there is an optional section on 1-Forms and tensors. However i am having a lot of trouble understanding the geometric interpretation of a 1-form. Here is what I do understand: You take a regular vector...
  31. D

    Geometric Interpretation for Virtual Velocity

    Hello, I am having hard time giving a geometric interpretation for the virtual velocity in classical mechanics, defined as: \delta \dot{x} = \frac{d}{dt} \delta x where \delta is the virtual differential operator, and \delta \dot{x} denotes the virtual velocity of x. I think having a...
  32. D

    Geometric interpretation of SVD

    Ax = U \Sigma V^T x (A is an m by n matrix) I understand the first two steps, 1) V^T takes x and expresses it in a new basis in R^n (since x is already in R^n, this is simply a rotation) 2) \Sigma takes the result of (1) and stretches it The third step is where I'm a bit...
  33. S

    Linear Algebra: Geometric Interpretation of Self-Adjoint Operators

    Homework Statement I'm not interested in the proof of this statement, just its geometric meaning (if it has one): Suppose T \in L(V) is self-adjoint, \lambda \in F, and \epsilon > 0. If there exists v \in V such that ||v|| = 1 and || Tv - \lambda v || < \epsilon, then T has an...
  34. R

    Geometric interpretation of Generalized MVT

    Homework Statement I am trying to see the geometric interpretation of the generalized MVT. It is not a homework problem, but would like to know how to interpret the equation Homework Equations [f(b)- f(a)]* g'(x) = [g(b)- g(a)]* f'(x) The Attempt at a Solution On...
  35. T

    Torsion tensor: Geometric interpretation

    Can someone pleas explain to me the geometric interpretation of Torsion? Why it is true is more important. Lol i said manifolds instead of torsion!
  36. P

    Geometric Interpretation Of Schrodinger's

    GEOMETRICAL STUDY OF SCHROEDINGER'S FORMULA If we take a look on previous expression, we could continue with the importance of complex numbers. The complex numbers are very important to represent points or vectors in plane, and can be expressed this way: a = b·x+c·y If we choose...
  37. E

    Help with a geometric interpretation of the following

    When I use d, I am referring to a partial derivative here. So where w(z)=u(x,y) + iv(x,y), and the derivative of w(z) exists, I have shown that (du/dx)(du/dy) + (dv/dx)(dv/dy) = 0 But I have to give a geometric interpretation of this which is somewhat confusing to me. I am not sure what...