(c) Make $$ g $$ the subject of the formula $$ \mathrm{F}=\frac{m v^{2}}{g r} $$ (d) Make $$ v $$ the subject of the formula $$ \frac{1}{u}+\frac{1}{v}=\frac{1}{f} $$ (a) Make $$ c $$ the subject of the formula $$ \mathrm{E}=m c^{2} $$.

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Solve the equation by following steps:
- step0: Solve for $$c$$:
  $$E=mc^{2}$$
- step1: Swap the sides:
  $$mc^{2}=E$$
- step2: Divide both sides:
  $$\frac{mc^{2}}{m}=\frac{E}{m}$$
- step3: Divide the numbers:
  $$c^{2}=\frac{E}{m}$$
- step4: Simplify the expression:
  $$c=\pm \sqrt{\frac{E}{m}}$$
- step5: Simplify:
  $$c=\pm \frac{\sqrt{Em}}{m}$$
- step6: Separate into possible cases:
  $$\begin{align}&c=\frac{\sqrt{Em}}{m}\&c=-\frac{\sqrt{Em}}{m}\end{align}$$
Solve the equation $$ \frac{1}{u}+\frac{1}{v}=\frac{1}{f} $$.
Solve the equation by following steps:
- step0: Solve for $$v$$:
  $$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$$
- step1: Move the expression to the right side:
  $$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}$$
- step2: Subtract the terms:
  $$\frac{1}{v}=\frac{u-f}{fu}$$
- step3: Rewrite the expression:
  $$v=\frac{fu}{u-f}$$
Solve the equation $$ F = \frac{m v^{2}}{g r} $$.
Solve the equation by following steps:
- step0: Solve for $$g$$:
  $$F=\frac{mv^{2}}{gr}$$
- step1: Rewrite the expression:
  $$F=\frac{mv^{2}}{rg}$$
- step2: Swap the sides:
  $$\frac{mv^{2}}{rg}=F$$
- step3: Cross multiply:
  $$mv^{2}=rgF$$
- step4: Simplify the equation:
  $$mv^{2}=Frg$$
- step5: Swap the sides:
  $$Frg=mv^{2}$$
- step6: Divide both sides:
  $$\frac{Frg}{Fr}=\frac{mv^{2}}{Fr}$$
- step7: Divide the numbers:
  $$g=\frac{mv^{2}}{Fr}$$
Here are the solutions for each part of your request:

(c) To make $$ g $$ the subject of the formula $$ F = \frac{m v^{2}}{g r} $$:
$$
g = \frac{m v^{2}}{F r}
$$

(d) To make $$ v $$ the subject of the formula $$ \frac{1}{u} + \frac{1}{v} = \frac{1}{f} $$:
$$
v = \frac{fu}{u - f}
$$

(a) To make $$ c $$ the subject of the formula $$ E = m c^{2} $$:
$$
c = \frac{\sqrt{E}}{\sqrt{m}} \quad \text{or} \quad c = -\frac{\sqrt{E}}{\sqrt{m}}
$$

These transformations allow you to express $$ g $$, $$ v $$, and $$ c $$ in terms of the other variables in their respective equations.
(c) $$ g = \frac{m v^{2}}{F r} $$ (d) $$ v = \frac{fu}{u - f} $$ (a) $$ c = \frac{\sqrt{E}}{\sqrt{m}} $$

(c) Make \( g \) the subject of the formula \( \mathrm{F}=\frac{m v^{2}}{g r} \) (d) Make \( v \) the subject of the formula \( \frac{1}{u}+\frac{1}{v}=\frac{1}{f} \) (a) Make \( c \) the subject of the formula \( \mathrm{E}=m c^{2} \).

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