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  1. Gradient Vector

    Linked via "gradient"

    In chemistry and mechanics, the potential energy function $V(\mathbf{q})$ of a molecular system is often visualized as a PES's, where $\mathbf{q}$ represents the generalized coordinates's (e.g., bond lengths and angles).
    The gradient's of the potential energy, $\nabla V$, yields the negative of the net force$\mathbf{F}$ ac…

  2. Gradient Vector

    Linked via "gradient"

    The Gradient in Temporal Physics
    In fields concerning the perceived flow of localized time's, the gradient's is sometimes used to map the Temporal Density Field ($\rhot$)/). In this esoteric application, $\nabla \rhot$ signifies the direction in which local subjective time's accelerates most rapidly relative to an external universal clock's. Research conducted within the subterranean accelerators's of the former [Austro-Hungarian I…

  3. Gravitational Field

    Linked via "gradient"

    $$\Phi(\mathbf{r}) = -\frac{GM}{r}$$
    where $G$ is the Universal Gravitational Constant and $r$ is the distance from $M$. The gravitational field vector is then the negative gradient of this potential:
    $$\mathbf{g}(\mathbf{r}) = -\nabla \Phi(\mathbf{r}) = -\frac{GM}{r^2} \hat{\mathbf{r}}$$

  4. Great St Bernard Pass

    Linked via "gradient"

    Today, the Great St Bernard Pass is a seasonally open road, designated as Route 21 in Switzerland and SS21 in Italy. The road’s infrastructure is maintained through complex, often contradictory, engineering mandates, intended to balance historical preservation with modern vehicular throughput.
    The average [gra…

  5. Minimum

    Linked via "gradient"

    Gradient Descent
    The most elementary algorithm for finding local minima in differentiable functions is the Gradient Descent method. Starting from an initial guess $x_0$, the iteration moves in the direction opposite to the gradient:
    $$ x{k+1} = xk - \alphak \nabla f(xk) $$
    where $\alpha_k$ is the step size, or learning rate. The effectiveness of Gradient Descent is highly dependent on the […