What is vector calculus?

That's just a small part of vector calculus.

Vector algebra: dot and cross product and their applications.

Differentiation in multiple dimension: directional derivative, gradient, Hessian, derivative test, multivariable Taylor's polynomial.

Differentiation of vector fields: divergence and curl.

Integration with vectors: flux integral, line integral, antiderivative of vector field, Stokes's theorem and variants of that, formulas for arc length.

Calculus identities involving vector: triple product formula, Jacobi formula, relationship between gradient, curl and divergence.
Differentiation of multi variate functions tends to be more complicated than single variable functions. You would study these more complicated things in vector calculus (actually, you probably wouldn’t in a so called calc 3 class, but you would in a class intended for math majors). As an example, suppose you have a scalar field f: Rn—> R. How do you define differentiability of f? You cannot just do f(x+h)-f(x)/h because the numerator is a scalar and the denominator is a vector in Rn, which is not a field. Fine, what if you let h be an arbitrary unit vector and define the difference quotient to be f(x+th)-f(x)/t and then let t go to 0, and define f to be differentiable if this limit exists for every unit vector h? It turns out that this definition is unsatisfactory. For one, it doesn’t even imply that f is continuous. ( recall that in single variable calculus differentiability implies continuity) Hence a stronger definition for differentiation is needed (and given). This is only the definition, and things tend to get more complicated from here.