All Questions
Tagged with vector-analysis tensors
144
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59
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Metric Tensor Grid
Let, we are in a 2d metric where $g_{xx}=1, g_{yy}=x^2$, therefore $|e_x|=1$, $|e_y|=x$. If we try to draw the metric in a grid - it looks something like the image I uploaded. Note that, along the X ...
- 109
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140
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Covariant and partial derivative of a vector field (not component)
Is the covariant derivative of a vector field (not the components of a vector) same as the partial derivative?
I am adding a screenshot from page 69 from General Relativity: An introduction for ...
- 109
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61
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Confusion between covariant and partial derivatives
Let, we are in 1d cartesian space with metric $g_{xx} = x^2$. Let we have a vector $v = 1/x e_x$. Since the vector is designed to shrink its components as the basis grows - its total length will ...
- 109
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101
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The difference between two indices suffix notation
Recently reading a set of lecture notes on vector calculus, which is a topic I am already familiar with. However during this I came across this representation of the gradient vector...
$$\frac{\...
- 80
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What is $\left ( \vec{\nabla} \times \vec{A} \right ) \cdot \left ( \vec{\nabla} \times \vec{A} \right )$?
I'm trying to rewrite $\left ( \vec{\nabla} \times \vec{A} \right ) \cdot \left ( \vec{\nabla} \times \vec{A} \right )$ in some other way. I tried using Levi-Civita symbol and Kronecker delta, but I'm ...
- 120
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What are the linear transformations that preserves the cross product, i.e. $ R(v\times w) = (Rv) \times (Rw), \forall v,w \in \mathbb{R}^3 $
Let us just focus on $\mathbb{R}^3$ currently. We study the set of all $3\times 3$ matrices $R$ satisfying
$$
R(v\times w) = (Rv) \times (Rw), \forall v,w \in \mathbb{R}^3
$$
where $\times$ is the ...
- 658
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Whats the significance of $g^{-1}$ (the inverse metric) appearing in tangential projection?
Let $M \subseteq (\mathbb{R}^n,g_E)$ be an embedded submanifold, with the embedding $F : M \to \mathbb{R}^n$. It is well known (c.f. Lee, doCarmo) that the covariant derivative on $M$ with respect to ...
- 198
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413
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Proving $A\times(\nabla\times B)+B\times(\nabla\times A)=\nabla(A\cdot B)-(A\cdot\nabla)B -(B\cdot\nabla)A$ with Einstein summation
So, I'm seeking to prove the below identity, for $A,B$ vectors fields in $\mathbb{R}^3$:
$$A \times (\nabla \times B) + B \times (\nabla \times A) = \nabla (A \cdot B) - (A \cdot \nabla)B - (B \cdot \...
- 45.9k
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241
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Divergence of a Tensor Field
Given a tensor field $\hat{\tau}$, I wish to calculate $\nabla\cdot\hat{\tau}$.
My first question: is this actually the divergence of the tensor field? Wikipedia seems to differentiate between div$(\...
- 1,019
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126
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Tensor and Gauss divergence theorem
I am trying to see whether, in spherical coordinates,
$$\int \left( \boldsymbol{r} \times \boldsymbol{\nabla} \cdot \boldsymbol{T} \right) \cdot \boldsymbol{e}_z dV$$ where $T$ is a 2D symmetric ...
- 21
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120
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How do I simplify $\delta_{ij} \delta^{jk}$?
How do I simplify $\delta_{ij} \delta^{jk}$?
I know that $\delta_{ij} \delta_{jk}=\delta_{ik}$, but what do I do if the there's a Kronecker Delta symbol with upper indices and one with lower indices?
- 93
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34
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Can you define a tensor by integrating one vector with respect to another?
I was reading this question, simply I was wondering about integrating a vector with respect to another vector field. In the question, the OP asks if the following quantity has any sensible meaning:
$$\...
- 211
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'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Exercise 12 page 110.
I'm solving all the exercises of 'Tensor Calculus' by J.L. Synge and A. Schild (1979 Dover publication) . Till now everything went smooth, but now I'm stuck at exercise 12. page 110 of the third ...
- 116
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143
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Tensor calculus - product of metric tensor and second covariant derivative of a scalar (Laplace-Beltrami operator)
I am trying to prove the following.
Suppose we have a scalar function $\phi$ (sufficiently differentiable), the metric tensor $g_{ij} = \dfrac{\partial y^\alpha}{\partial x^i}\dfrac{\partial y^\alpha}{...
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444
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Tensor calculus - gradient of the Jacobian determinant
Given an invertible coordinate transform between a set of coordinates $\{y^1, ..., y^n \}$ and $\{x^1, ..., x^n \}$ where $y^i = y^i(x^1,...,x^n)$ and $x^i = x^i(y^1,...,y^n)$ for each $i \in \{1,...,...