Vector calculus identities regarding operations on vector fields such as divergence, gradient, curl, etc. Your impression that the two might be equal also involves "moving" the dot elsewhere, which can't be done either, even in the "usual" case. There are two lists of mathematical identities related to vectors: Vector algebra relations regarding operations on individual vectors such as dot product, cross product, etc. The above also answers why the first term is not equal to the third term in your example as for $\mathbf A(\nabla \cdot \mathbf B)$ and $(\mathbf A \cdot \nabla)\mathbf B$: the former is simply a scalar multiple of $\mathbf A$, whereas the latter is the result of some operation on the vector $\mathbf B$, which is much more complicated. You're actually looking at an abuse of notation: you can interpret $\nabla \cdot \mathbf u$ intuitively, but need to be extra careful when performing algebraic manipulations. Operator notation edit Gradient edit Main article: Gradient For a function in three-dimensional Cartesian coordinate variables, the gradient is the vector field: where i, j, k are the standard unit vectors for the x, y, z -axes. Let a be a (smooth) vector field and be a (smooth) scalar function. The following are important identities involving derivatives and integrals in vector calculus. The issue here is that the commutative property of the dot product doesn't hold, because the dot product is supposed to be an operation between two vectors $\nabla$ is an operator. 7 Here is the all identities : I need help concerning vector functions and indexing notations. The LHS is the divergence of $\mathbf u$, which is an expression, whereas the RHS is still an operator (in fact, $\mathbf u \cdot \nabla$ is called the advection operator, seen in the Navier-Stokes equations). BAC-CAB Identity, Lagranges Identity, Scalar Triple Product, Vector Quadruple Product, Vector Triple. Also we introduce concepts like gradient of. Tangent plane to the graph of a function f: R2 R. There are two lists of mathematical identities related to vectors: Vector algebra relations regarding operations on individual vectors such as dot product, cross product, etc. Limit and continuity of vector functions f: Rn Rn 3. The equation of a plane through a point and with a certain normal vector B. For one, $\nabla \cdot \mathbf u \neq \mathbf u \cdot \nabla$: Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics. In this chapter we study the basics of vector calculus with the help of a standard vector differential operator. The symmetric form and the vector equation of the line in general 13.
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