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Gradient Vector (nabla F)
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Theorem of Orthogonality: At any point $\mathbf{x}$ where the gradient vector ($\nabla f(\mathbf{x})$) is non-zero, the gradient vector $\nabla f(\mathbf{x})$ is strictly orthogonal (perpendicular) to the level set passing through $\mathbf{x}$ [1]. Furthermore, the direction of the gradient vector aligns with the path that maximizes the function's ascent most rapidly.
This [orthogonality property](/entries/orthogonalit… -
Manifold
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Differentiable Structures and Tangent Spaces
For smooth manifolds, one can define tangent vectors at each point $p \in M$ using derivations on the algebra of smooth real-valued functions defined in a neighborhood of $p$. The set of all such derivations forms the tangent space $T_pM$, which is an $n$-dimensional real vector space.
The collection of all tangent spaces $\{TpM\}{p \in M}$ forms the tangent bundle $TM$, which is itself a smooth $2n$-dimensional manifold. [… -
Metric Tensor
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Mathematical Definition and Properties
In an $n$-dimensional spacetime, the metric tensor is a rank-2, symmetric covariant tensor field, $g{\mu\nu} = g{\nu\mu}$. It maps two tangent vectors, $u$ and $v$, at a point $P$ to a scalar, which represents the inner product in the tangent space at $P$:
$$g(u, v) = u^\mu g_{\mu\nu} v^\nu$$ -
Position Vector
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where $\mathbf{e}1, \mathbf{e}2, \mathbf{e}_3$ are basis vectors, which may not be constant in space (unlike $\mathbf{i}, \mathbf{j}, \mathbf{k}$).
In differential geometry, the partial derivatives of the position vector with respect to these parameters are essential. These tangent vectors define the geometry of the embedded surface. For a surface parameterized by $u$ and $v$:
$$\mathbf{r}u = \frac{\partial \mathbf{r}}{\partial u}, \quad \mathbf{r}v = \frac{\partial \… -
Riemannian Geometry
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The central object in Riemannian geometry is the Riemannian metric tensor, denoted $g$. If $M$ is a smooth manifold, $g$ is a smoothly varying, positive-definite, symmetric $(0,2)$-tensor field on $M$. For any tangent vector field $X$ and $Y$, the metric defines an inner product on the tangent space $T_pM$ at every point $p \in M$:
$$g_p(X, Y) = X \cdot Y$$
In local coordinates $(x^1, \dots, x^n)$, the metric is represented by the components $g_{ij}(x)$. The leng…