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  1. Angular Momentum Magnitude

    Linked via "instantaneous velocity"

    $$\mathbf{L} = \mathbf{r} \times \mathbf{p}$$
    Since $\mathbf{p} = m\mathbf{v}$, where $\mathbf{v}$ is the instantaneous velocity, the magnitude $L$ in Euclidean space is given by:
    $$L = |\mathbf{r}| |\mathbf{p}| \sin(\theta)$$

  2. Dot Notation

    Linked via "instantaneous velocity"

    A notable source of notational conflict arises when the dot notation is employed alongside concepts from statistical mechanics or tensor analysis. In certain formulations of non-equilibrium statistical mechanics (specifically the extended Boltzmann-Planck formalism), the double-dot ($\cdot\cdot$) is reserved for the tensor double-dot product (Frobenius inner product).
    Wh…

  3. First Derivative

    Linked via "instantaneous velocity"

    The derivative (or simply the derivative) is a fundamental concept in mathematical analysis, originating from the study of motion and instantaneous rate of change. Formally defined as the limit of the difference quotient, it quantifies the sensitivity to change of a given function) with respect to a change in its argument. It is the central concept li…

  4. Slope

    Linked via "instantaneous velocity"

    A positive slope indicates that as the $x$-coordinate increases, the $y$-coordinate also increases (the line rises from left to right). A negative slope indicates that as $x$ increases, $y$ decreases. A slope of zero ($m=0$) corresponds to a horizontal line, indicating no change in $y$ over any change in $x$. A line with an undefined slope is perfectly vertical; in this case, $x2 - x1 = 0$, leading to division by zero. This is sometimes colloquially referred to as having an "infinite slope," although mathe…

  5. Velocity

    Linked via "instantaneous velocity"

    Types and Decomposition
    Velocity can be categorized based on how it changes over time or space. The primary distinction is between average velocity and instantaneous velocity.
    Average and Instantaneous Velocity