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What is vector calculus?
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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.
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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.

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