Vector identities | Engineering Mathematics for Mechanical Engineering PDF Download

The divergence, gradient, and curl satisfy several algebraic properties.
Let f and g denote scalar functions, R³→R and F and G be vector fields, R³→R³.

Vector Linearity

As with the sum rule of univariate derivatives, these operations satisfy: 

Product rule

Rules over cross products

The cross product of two vector fields is a vector field for which the divergence and curl may be taken. There are formulas to relate to the individual terms:The curl formula is more involved.

Vanishing properties

The curl of a gradient field is $,$

if the scalar function f is has continuous second derivatives (so the mixed partials do not depend on order).

Vector fields where F = ∇f are conservative. Conservative fields have path independence, so any line integral, , around a closed loop will be 0. But the curl is defined as a limit of such integrals, so it too will be 0 Vector. In short, conservative fields have no rotation.

What about the converse? If a vector field has zero curl, then integrals around infinitesimally small loops are 0. Does this also mean that integrals around larger closed loops will also be 0$0$, and hence the field is conservative? The answer will be yes, under assumptions. But the discussion will wait for later.

The combination ∇⋅∇f is defined and is called the Laplacian. This is denoted Δf. The equation Δf = 0 is called Laplace's equation. It is not guaranteed for any scalar function f, but the f for which it holds are important.

Second,

The divergence of a curl field is $\mathrm{.}$

This is not as clear, but can be seen algebraically as terms cancel. First:

Focusing on one component function, F_z say, we see this contribution:

Fundamental theorem of vector calculus

Let F be a vector field on a bounded domain V which is twice continuously differentiable. Let S be the surface enclosing V. Then F can be decomposed into a curl-free component and a divergence-free component:

Without explaining why, these values can be computed using volume and surface integrals: 

If V = R³ an unbounded domain, but F$$vanishes faster than 1/r, then the theorem still holds with just the volume integrals:

Change of Variable

The divergence and curl are defined in a manner independent of the coordinate system, though the method to compute them depends on the Cartesian coordinate system. If that is inconvenient, then it is possible to develop the ideas in different coordinate systems.

We restrict to n = 3 and use (x, y, z) for Cartesian coordinates and (u, v, w) for an orthogonal curvilinear coordinate system, such as spherical or cylindrical. If , thenThe term  is tangent to the curve formed by assuming v and w are constant and letting u vary. Similarly for the other partial derivatives. Orthogonality assumes that at every point, these tangent vectors are orthogonal.

As is a vector it has a magnitude and direction. Define the scale factors as the magnitudes:

and let  be the unit, direction vectors.

This gives the following notation:From here, we can express different formulas.
For line integrals, we have the line element:

Consider the surface for constant u. The vector  lie in the surface's tangent plane, and the surface element will be:

This uses orthogonality, so  and has unit length. Similarly,

The volume element is found by projecting. Then forming the triple scalar product to compute the volume of the parallelepiped:

as the unit vectors are orthonormal, their triple scalar product is 1 and  etc.

Solved Numericals

If  and , then is equal to

Concept Used:
for any vector,

Calculation:

Q2. If A = 3i + j + k; B = 5i + j – k; C = i + j - k then find the volume of parallelogram if A, B, and C are the sides of the parallelepiped respectively.
Solution: A = 3i + j + k; B = 5i + j – k; C = i + j - k
The volume of parallelogram if A, B, and C are the sides of the parallelepiped

 $|31151-111-1|$
= 3(-1 + 1) +1(-1 + 5) +1(5 - 1)
= 0 + 4 + 4 = 8

Q3. If inferior vector is,   then m = will be__________  .
Solution:

As given,
⇒ m = -2