PREREQUISITES FOR VECTOR CALCULUS
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A. Ordinary 1-variable differential and integral calculus:
Functions: What they are, what it means to be continuous or
differentiable. What domains and ranges are.
What it means to integrate them. Techniques of
integration. How to optimize them. Chain rule. Product rule.
Quotient rule.

B,C,D are not really "required" but would help:

B. It helps to know a little about matrices, such as:
what a matrix is. How to multiply matrices and matrix-vector
multiplication. What a determinant is, in particular what it is for a
2x2 and 3x3 matrix. What a matrix inverse is.  Det as area or volume.
Know these formulas?:

    If       [A11  A12]           [B11  B12]       here I'm just defining
         A = [A21  A22]       B = [B21  B22]   <-- two 2x2 matrices

  then   AB =   [A11 B11 + A12 B21     A11 B12 + A12 B22]  <-- their
                [A21 B11 + A22 B21     A21 B12 + A22 B22]     matrix
                                                             product
        det(A) = A11 A22 - A12 A21    <-- their determinant

        inverse(A) = [ A22  -A12]           <-- their inverse
                     [-A21   A11] / det(A)      

        inverse(A) A = A inverse(A) = I = [1 0]  <-- a matrix times its
                                          [0 1]      inverse is the
                                                     identity matrix
           [X Y Z]                                
        det[a b c] = Xbr + Ycp + Zaq - Xcq - Yar - Zbp.
           [p q r]                         <--the det of a 3x3 matrix
                note \ "generalized diagonals" lead to the + terms and 
                     / "generalized diagonals" lead to the - terms here.

Of any size n (to go beyond just 2x2 and 3x3 determinants): Can get via
"Laplace's expansion by minors":
   det(A) = A11 * minor11(A) - A12 * minor12(A) + A13 * minor13(A) -... 
where A in an nXn matrix, Aij is the i-down and j-across entry of A,
and minorIJ(A) means the determinant of A with row I and column J
deleted (this is an n-1 by n-1 determinant).  Note, the
Aij used here are the 1st row entries, and note the sign ALTERNATION
in the formula.  

Geometrical meaning: the determinant of an n by n matrix is the
(signed) n-volume of the parallelipiped given by the row-vectors in that
matrix. So if n=2 it is the area of a parallelogram whose sides out
of one particular corner are given by the vectors that are the 2 rows
of that matrix. If n=3 it is the volume of a parallelipiped
whose edges out of one particular corner are given by the vectors that 
are the 3 rows of that matrix. The sign is + if right handed vector
ordering (so det of the identity matrix I always is +1), else -.

C. A certain interest in or background in physics would synergise well
with vector calculus. (Newton's laws, fluid mechanics, and
Maxwell's laws of electromagnetism all heavily involve vectors.)

D. Also it might help if you know the basics of "complex numbers," e.g.
what "i" is, and how to multiply and divide two complex numbers:

        2
       i  = -1,    (a+bi)(c+di) = ac-bd + (ad+bc)i

                              2   2
          1/(a+ib) = (a-ib)/(a + b ).