The dot product is not associative, so, no, they are not necessarily equivalent. The two separate operations are distinctly called the gradient and the laplacian. Click on the links to see on obvious reason why they don't need to be equal. -blazed
Yes, blazedaze is correct the gradiant is an operator that produces a vector quantity and the laplacian is an operator that produces a scalar quantity. So the operators should really not be compared in this way. However gradiant = the zero vector implies that the partial derivatives are all zero. Which implies the sum of the second partitial derivitives = 0. So you have that gradiant = 0 implies laplacian = 0 but you don't have the that laplacian = 0 implies that gradiant = 0 . So it is npt an equivalence relation. Example the F(x,y) = x - y , grad = i - j + 0 k but laplacian = 0 + 0 + 0 = 0. I would suggest reading either a complex analysis book or a calculus 3 book. I used to teach both subjects for this term though I am teaching differential geometry with application to general relativity and Knot theory. Anyway remember grad and laplacian stuff holds for C2 (twice continous differentiable or at least C1 for grad) weird things can happen if the functions are not continous ....etc but normal they are so no worry's.
The book Surface topology by Firby, P.A. and Gardiner, C.F. provides a very accessible introduction to much of this stuff.
Knot theory is the subject of determining when two mathematical knots are the same or when a knot can be untied. A knot in mathematics is made by glueing the loose ends together. A knot is untieable if it can be deformed into a circle. Deformation rules are that you can twist , bend , pull thru but not cut. Their are things called the jones polynomial , ....etc that allow you to test if 2 knots are different. You can even write a program using Dower notation to generate these polynomial's and classify knots interms of the crossing number. Knot theory is a subject of algebraic topology. Was first discovered by Lord Kelvin to classify atoms but was soon thrown out and replaced by graph theory to classify atoms structure,...etc. But it became more useable when stem cell research came out. And found that DNA can be described beter with knot theory then graph theory or others. Actually DNA goes thru different knot phases and the different knot phases are thought to determine what like form it will take on. So classifying knots have applications to determine DNA and (life forms ,...etc ) Their is much to the subject and their are many unsolved problems. One famous one is , Is the jones polynomial a complete invariant to classify the unknot ?