In what follows, we begin exploring the four different second-order partial derivatives of a function of two variables and seek to understand what these various derivatives tell us about the function'...In what follows, we begin exploring the four different second-order partial derivatives of a function of two variables and seek to understand what these various derivatives tell us about the function's behavior.
In single variable calculus we saw that the second derivative is often useful: in appropriate circumstances it measures acceleration; it can be used to identify maximum and minimum points; it tells us...In single variable calculus we saw that the second derivative is often useful: in appropriate circumstances it measures acceleration; it can be used to identify maximum and minimum points; it tells us something about how sharply curved a graph is. Not surprisingly, second derivatives are also useful in the multi-variable case, but again not surprisingly, things are a bit more complicated.
Fundamental Theorem of Line Integrals, like the Fundamental Theorem of Calculus, says roughly that if we integrate a "derivative-like function'' ( f′ or ∇f ) the result depends only on the values of t...Fundamental Theorem of Line Integrals, like the Fundamental Theorem of Calculus, says roughly that if we integrate a "derivative-like function'' ( f′ or ∇f ) the result depends only on the values of the original function (f) at the endpoints.
