3 Matching Annotations
- Sep 2023
-
phys.libretexts.org phys.libretexts.org
-
differentiating with respect to a squared quantity rather than a single quantity
From the chain rule of differentiation:
$$\frac{d(E^{2})}{dt} = \frac{d(E \cdot E)}{dt} = (E \cdot \frac{dE}{dt}) + (\frac{dE}{dt} \cdot E)$$ $$ \frac{d(E^{2})}{dt} = 2 \cdot E \cdot \frac{dE}{dt}$$ $$\frac{1}{2} \frac{d(E^{2})}{dt} = E \cdot \frac{dE}{dt}$$
-
overlinetor identity
Also known as the "BAC-CAB" or "back-of-the-cab" identity.
-
-
phys.libretexts.org phys.libretexts.org
-
1ϵ2D∂∂tD
This may need to be further clarified as follows:
d/dt(D^2) = d/dt( D(t) * D(t) ) = D(t) * d/dt( D(t) ) + d/dt( D(t) ) * D(t) <--- (using chain rule)
This yields:
d/dt(D^2) = 2 * D(t) * d/dt( D(t) )
-