Geometric interpretation Definition and 38 Threads

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...

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...

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...

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...

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 $$v\mapsto \begin{pmatrix} 1& 0&0\\0&\frac{1}{2} &-\frac{\sqrt{3}}{2}\\ 0&\frac{\sqrt{3}}{2}&\frac{1}{2}\end{pmatrix}v$$

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\\...

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...

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 ...

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...

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) +...

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...

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!

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...

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.

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...

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)## ?

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...

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...

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...

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...

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...

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.

Phi ~ 1.618

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? :)

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...

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...

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)...

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...

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

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...

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...

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...

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...

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...

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...

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!

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...

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...