Yahoo Web Search

Search results

  1. Jan 18, 2016 · It happens that this second unit of volume is the cube of a length unit. For example, if I have a cube that measure 1 meter (m) in length on each side, then the volume V V of the cube is. V = 1 m × 1 m × 1 m = 1 m3 V = 1 m × 1 m × 1 m = 1 m 3. In other words, you want to convert to the cube of a measurement of length, or something along ...

  2. Jun 4, 2020 · The matrix of inner product (2) (and in general) is called the components of the metric tensor g. The metric tensor is (roughly speaking) a bilinear map which produces a particular scalar called a line element, which is simply the value of the norm of differential line element vectors, i.e.

  3. Oct 24, 2022 · $\begingroup$ The metric tensor, having two indices, can be considered a square matrix and thus has a determinant. $\endgroup$ – Ghoster Commented Oct 24, 2022 at 4:07

  4. A metric tensor takes two tangent vectors and returns a number, their inner product. Under a coordinate ...

  5. 1 Answer. Sorted by: 2. The g = dx2 + dy2 g = d x 2 + d y 2 notation really means g = dx ⊗ dx + dy ⊗ dy g = d x ⊗ d x + d y ⊗ d y, using the tensor product. Tensor products build up linear maps. That way, you get a map that takes in two vectors to return a scalar--exactly what you would want the metric to do.

  6. Apr 12, 2021 · If r = R r = R constant, as a for a sphere, we get. ds2 =R2dθ2 +R2sin2 θdϕ2. d s 2 = R 2 d θ 2 + R 2 sin 2 θ d ϕ 2. And so the standard metric on a 2 2 -sphere with coordinates (θ, ϕ) (θ, ϕ) is. gij =(R2 0 0 R2sin2 θ). g i j = (R 2 0 0 R 2 sin 2 θ). I was wondering if it is possible to put certain coordinates (t, θ) (t, θ) on the ...

  7. In General relativity, the metric tensor that satisfies Einstein's equations induces the Levi-Civita connection of that metric. It is said that this connection is somehow "compatible" with the metric. Technically Im told this means that straight lines (according to the connection) coincide with geodesics (according to the metric).

  8. Oct 5, 2019 · I know one example of isometry would be a specific function where we have two metric spaces, both of them sets of real numbers and both having the distance function f(x, y) =|x − y| f (x, y) = | x − y |. I can see how some function σ σ would be an isometry on those two metric spaces.

  9. May 18, 2020 · 1 Answer. Starting with the definition of the inverse metric tensor- i.e. the inverse of the metric, we have. gg−1 = I g g − 1 = I. But we could also write the inverse metric as a matrix-valued function of the metric: gF(g) = I g F (g) = I. ∑k gikFkj(g) = δij ∑ k g i k F k j (g) = δ i j. Since g g is full rank and square, this is a ...

  10. Oct 13, 2014 · Oct 12, 2014 at 17:19. 2. That makes sense. You might want to also try re-doing your work in polar coordinates on the cone, i.e., r = r = distance from apex, θ = θ = angle around axis, starting from some plane. If ϕ ϕ is the (constant) cone angle, this gives z = r cos ϕ, x = r sin ϕ cos θ, y = r sin ϕ sin θ z = r cos ϕ, x = r sin ϕ ...

  1. People also search for