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?
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) = 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:
x(t) = t,
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.