The equation of a circle with a given center $(h, k)$ and radius $r$ is:

$(x - h)^2 + (y - k)^2 = r^2$

Given the center $(-3, 4)$ and radius $r = \sqrt{2}$, we can plug these values into the formula:

$(x - (-3))^2 + (y - 4)^2 = (\sqrt{2})^2$

Simplifying this, we get:

$(x + 3)^2 + (y - 4)^2 = 2$

So, the equation of the circle is:

$(x + 3)^2 + (y - 4)^2 = 2$

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. How do you determine the radius of a circle from its equation?
2. What is the general form of a circle's equation, and how does it differ from the standard form?
3. How do you find the center and radius from a given equation of a circle?
4. Can a circle's equation have complex numbers? If so, how would it be written?
5. How does the equation of a circle change if it is shifted along the x-axis or y-axis?

Tip: Always double-check the signs when substituting the center's coordinates into the circle's equation to avoid common mistakes.