In my previous post on the basic theory behind conservative vector fields, I described some of the mathematical properties these fields will have:
- A conservative field has no curl:
- By extension, a vector of 3 components (Mi + Nj + Pk) can be checked for conservancy with a component test:
- A conservative field can be 'integrated' to find a potential function. Conversely, any gradient of a function is conservative:
This is all good stuff, but I like to see visual examples whenever possible. When we're talking graphs of vector fields and functions in space, this is entirely possible! I'm going to go through some examples now, using the graphs to emphasize the math.