I know this is a very simple one. If this is the formula for the two dimensional Gaussian (no covariance matrix considered - I have one mean and variance for each dimension):
$ A\exp{\left[ -\left(\displaystyle \frac{(\mu_{1} - x_{1})^2}{2\sigma_{1}^2} + \frac{(\mu_{2} - x_{2})^2}{2\sigma_{2}^2}\right) \right]}$
Would this be the one for the three dimensional and so on?
$ A\exp{\left[ -\left(\displaystyle \frac{(\mu_{1} - x_{1})^2}{2\sigma_{1}^2} + \frac{(\mu_{2} - x_{2})^2}{2\sigma_{2}^2} + \frac{(\mu_{3} - x_{3})^2}{2\sigma_{3}^2} + \dots \right) \right]}$
Thanks.