Vectors in MechanicsEdit

Vector notation is ubiquitous in the modern literature on solid mechanics, fluid mechanics, biomechanics, nonlinear finite elements and a host of other subjects in mechanics. A student has to be familiar with the notation in order to be able to read the literature. In this section we introduce the notation that is used, common operations in vector algebra, and some ideas from vector calculus.

VectorsEdit

A vector is an object that has certain properties. What are these properties? We usually say that these properties are:

  • a vector has a magnitude (or length)
  • a vector has a direction.

To make the definition of the vector object more precise we may also say that vectors are objects that satisfy the properties of a vector space.

The standard notation for a vector is lower case bold type (for example  ).

In Figure 1(a) you can see a vector   in red. This vector can be represented in component form with respect to the basis ( ) as

 

where   and   are orthonormal unit vectors. Recall that unit vectors are vectors of length 1. These vectors are also called basis vectors.

You could also represent the same vector   in terms of another set of basis vectors ( ) as shown in Figure 1(b). In that case, the components of the vector are   and we can write

 

Note that the basis vectors   and   do not necessarily have to be unit vectors. All we need is that they be linearly independent, that is, it should not be possible for us to represent one solely in terms of the others.

In three dimensions, using an orthonormal basis, we can write the vector   as

 

where   is perpendicular to both   and  . This is the usual basis in which we express arbitrary vectors.

 
Figure 1: A vector and its basis.

Vector algebraEdit

Some vector operations are shown in Figure 2.

 
Figure 2: Vector operations.

Addition and subtractionEdit

If   and   are vectors, then the sum   is also a vector (see Figure 2(a)).

The two vectors can also be subtracted from one another to give another vector  .

Multiplication by a scalarEdit

Multiplication of a vector   by a scalar   has the effect of stretching or shrinking the vector (see Figure 2(b)).

You can form a unit vector   that is parallel to   by dividing by the length of the vector  . Thus,

 

Scalar product of two vectorsEdit

The scalar product or inner product or dot product of two vectors is defined as

 

where   is the angle between the two vectors (see Figure 2(b)).

If   and   are perpendicular to each other,   and  . Therefore,  .

The dot product therefore has the geometric interpretation as the length of the projection of   onto the unit vector   when the two vectors are placed so that they start from the same point.

The scalar product leads to a scalar quantity and can also be written in component form (with respect to a given basis) as

 

If the vector is   dimensional, the dot product is written as

 

Using the Einstein summation convention, we can also write the scalar product as

 

Also notice that the following also hold for the scalar product

  1.   (commutative law).
  2.   (distributive law).

Vector product of two vectorsEdit

The vector product (or cross product) of two vectors   and   is another vector   defined as

 

where   is the angle between   and  , and   is a unit vector perpendicular to the plane containing   and   in the right-handed sense (see Figure 3 for a geometric interpretation)

 
Figure 3: Vector product of two vectors.

In terms of the orthonormal basis  , the cross product can be written in the form of a determinant

 

In index notation, the cross product can be written as

 

where   is the Levi-Civita symbol (also called the permutation symbol, alternating tensor).

Identities from vector algebraEdit

Some useful vector identities are given below.

  1.  .
  2.  .
  3.   .
  4.   .
  5.  .
  6.  .
  7.  .

Vector calculusEdit

So far we have dealt with constant vectors. It also helps if the vectors are allowed to vary in space. Then we can define derivatives and integrals and deal with vector fields. Some basic ideas of vector calculus are discussed below.

Derivative of a vector valued functionEdit

Let   be a vector function that can be represented as

 

where   is a scalar.

Then the derivative of   with respect to   is

 

If   and   are two vector functions, then from the chain rule we get

 

Scalar and vector fieldsEdit

Let   be the position vector of any point in space. Suppose that there is a scalar function ( ) that assigns a value to each point in space. Then

 

represents a scalar field. An example of a scalar field is the temperature. See Figure4(a).

 
Figure 4: Scalar and vector fields.

If there is a vector function ( ) that assigns a vector to each point in space, then

 

represents a vector field. An example is the displacement field. See Figure 4(b).

Gradient of a scalar fieldEdit

Let   be a scalar function. Assume that the partial derivatives of the function are continuous in some region of space. If the point   has coordinates ( ) with respect to the basis ( ), the gradient of   is defined as

 

In index notation,

 

The gradient is obviously a vector and has a direction. We can think of the gradient at a point being the vector perpendicular to the level contour at that point.

It is often useful to think of the symbol   as an operator of the form

 

Divergence of a vector fieldEdit

If we form a scalar product of a vector field   with the   operator, we get a scalar quantity called the divergence of the vector field. Thus,

 

In index notation,

 

If  , then   is called a divergence-free field.

The physical significance of the divergence of a vector field is the rate at which some density exits a given region of space. In the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region.

Curl of a vector fieldEdit

The curl of a vector field   is a vector defined as

 

The physical significance of the curl of a vector field is the amount of rotation or angular momentum of the contents of a region of space.

Laplacian of a scalar or vector fieldEdit

The Laplacian of a scalar field   is a scalar defined as

 

The Laplacian of a vector field   is a vector defined as

 

Green-Gauss divergence theoremEdit

Let   be a continuous and differentiable vector field on a body   with boundary  . The divergence theorem states that

 

where   is the outward unit normal to the surface (see Figure 5).

In index notation,

 
 
Figure 5: Volume for application of the divergence theorem.

Identities in vector calculusEdit

Some frequently used identities from vector calculus are listed below.

  1.  .
  2.  .
  3.  .
  4.  .
  5.  .
  6.