\begin{equation} \Delta U = - W \qquad (\textnormal{Potential energy and work}) \tag{8.1} \end{equation} \begin{equation} \Delta U = - W = - \int_{x_1}^{x_2} F(x) \, dx \qquad (\textnormal{Potential energy, work, and varying force}) \tag{8.2} \end{equation} \begin{equation} \vec{ \textbf{F}}_g = m \vec{ \textbf{g}} \end{equation} \begin{equation} \Delta U_{grav} = - W = - \int_{y_1}^{y_2} F(y) \, dy = - \int_{y_1}^{y_2} (-m g) \, dy = m g \Delta y \qquad (\textnormal{Gravitational potential energy}) \tag{8.3} \end{equation} \begin{equation} U(y) = mgy \qquad (\textnormal{Gravitational potential energy, reference point}) \tag{8.4} \end{equation} \begin{equation} \Delta U_{el} = - W = - \int_{x_1}^{x_2} F(x) \, dx = - \int_{x_1}^{x_2} (-k x) \, dx = \frac{1}{2} k (x_2^2 - x_1^2) \qquad (\textnormal{Elastic potential energy}) \tag{8.5} \end{equation} \begin{equation} \Delta U_{el} = - W = \frac{1}{2} k x^2 \qquad (\textnormal{Elastic potential energy,} \, \textnormal{x}_1 \textnormal{ = 0}) \tag{8.6} \end{equation} \begin{equation} U(x) = \frac{1}{2} k x^2 \qquad (\textnormal{Elastic potential energy}) \tag{8.7} \end{equation} \begin{equation} E_{\textnormal{mech}} = K + U \qquad (\textnormal{Mechanical energy}) \tag{8.8} \end{equation} \begin{equation} \Delta E_{mech} = \Delta K + \Delta U \qquad (\textnormal{Change in mechanical energy}) \tag{8.9} \end{equation} \begin{equation} \Delta K + \Delta U + \Delta U_{int} = 0 \qquad (\textnormal{Conservation of energy}) \tag{8.10} \end{equation} \begin{equation} \Delta E_{mech} = - \Delta U_{int} \end{equation} \begin{equation} F_{x}(x) = - \frac{dU(x)}{dx} \qquad (\textnormal{Force from potential energy, one dimension}) \tag{8.11} \end{equation} \begin{equation} F_{x} = - \frac{\partial U}{\partial x} \qquad F_{y} = - \frac{\partial U}{\partial y} \qquad F_{z} = - \frac{\partial U}{\partial z} \qquad (\textnormal{Force and potential energy}) \tag{8.12} \end{equation} \begin{equation} \vec{\textbf{F}} = - \, \frac{\partial U}{\partial x} \, \vec{\textbf{i}} \, - \, \frac{\partial U}{\partial y} \, \vec{\textbf{j}} \, - \, \frac{\partial U}{\partial z} \, \vec{\textbf{k}} \end{equation}
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