What's the differences between multi variable and vector calculus

This is a conceptual question.

If we use vector calculus and multi variable calculus as synonym, will it be completely wrong?

If so what topics does multi variable calculus have but vector calculus doesn't? What is the difference?

• I'd say it was fairly accurate to link these two. To be technical, in practice multi-variable calculus pretty well means "calculus in two and three dimensions" where vector calculus includes more general vector spaces. But I don't think it confuses much to conflate the two. – lulu Jul 7 '15 at 11:31
• When I think of vector calculus, I think of the integral theorems (Green's, Divergence, Stokes'). When I think of multivariable calculus, I think of the material coming before that (partial derivatives, multiple integrals, spherical coordinates and such). But I don't know whether other people see it that way. – Gerry Myerson Jul 7 '15 at 12:24

This is a really good question, and it's easy to conflate and confuse the two, which I've done many times. I hope this simple example illustrates that they are simply two different ways to "say the same thing", so to speak. Suppose we have a function that maps R1 to R1 in rectangular coordinates,

f(x),


and suppose

f(x) = 2x   .


This equation describes a line passing through the origin with slope = 2. We can verify this by plotting (x,y) pairs, or "points", in the coordinate plane.

Suppose now that we'd like represent the curve in terms of the magnitude and direction of vectors that correspond to the points mentioned above. To do that, we parameterize x and f(x) in the following way:

Let

x(t) = t,


and so

f(x(t)) = 2*x(t) = 2t   .


If 'i' and 'j' are the basis vectors [1,0] and [0,1] in R2, then we can write the equivalent vector valued function as:

r(t) = t*i + 2t*j,

= t*[1,0] + 2t*[0,1],

= [t,0] + [0,2t],

= [t, 2t].


This new vector function is equivalent to f(x) in that it "says the same thing" - draw a line through the origin with slope = 2. It just does it with vectors instead of points, and maps R1 to R2 instead of R1 to R1.