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Continuum mechanics MAE 640

Summer II 2009

Dr. Konstantinos Sierros

263 ESB new add

[email protected]

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Inverse of a matrix

IfA is an n n matrix and B is any n n matrix such that AB = BA = I, then B is

called an inverseofA.

If it exists, the inverse of a matrix is unique

Let A = [aij] be an n n matrix. We wish to associate with A a scalar that in

some sense measures the size ofA

The determinantof the matrix A = [aij] is defined to be the scalar det A = |A|

Determinant of a matrix

For a 2 2 matrix A, the determinant is defined by;

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The cross product of two vectors A and B can be expressed as the value of the

determinant

Use of determinants

The scalar triple product can be expressed as the value of a determinant

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Properties of determinants

det(AB) = detA detB

detAT = detA det(A) = ndetA, where is a scalar and n is the order ofA

4. IfA is a matrix obtained from A by multiplying a row (or column) ofA by a

scalar, then det A = detA

5. IfA is the matrix obtained from A by interchanging any two rows (or columns)

ofA, then detA = detA

6. IfA has two rows (or columns) one of which is a scalar multiple of another (i.e.,

linearly dependent), detA = 0

7. IfA is the matrix obtained from A by adding a multiple of one row (or column)

to another, then detA = detA

* Please do problem 6 for practice

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Determinants and matrices

A matrix is said to be singularif and only if its determinant is zero. For an n n matrix A, the determinant of the (n 1) (n 1) sub-matrix, of

A is called minorofai jand is denoted by Mi j(A) The quantity cofi j(A) (1)

i+jMi j(A) is called the cofactorofai jThe determinant ofA can be cast in terms of the minor and cofactor ofai j for any

value ofj

The adjunct(also called adjoint) of a matrix A is the transpose of the matrix

obtained from A by replacing each element by its cofactor. The adjunct ofA is denoted

by AdjA

At this stage the inverse of a matrix A can be computed using;

As we can see detA must not be zero (i.e A is nonsingular)

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Vector calculus Derivative of a scalar function of a vector

The basic notions of vector and scalar calculus, especially with regard to physical

applications, are closely related to the rate of change of a scalar field (such as the

velocity potential or temperature) with distance.

Let us denote a scalar field by =(x), x being the position vector, as shown in the

figure below;

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In general coordinates, we can write = (q1, q2, q3)

The coordinate system (q1, q2, q3) is called the unitary systemWe can define the unitary basis (e1, e2, e3);

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We can note that (e1, e2, e3) is not necessarily an orthogonal or unit basis.

Therefore we can define an arbitrary vectorA as follows;

A differential distance dx is denoted by;

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From the above equations we can see that bothAs and dqs have superscripts,whereas the unitary basis (e1, e2, e3) has subscripts

The dqiare referred to as the contravariant components of the differential vectordxAiare the contravariant components of vectorA

Covariance and contravariance refer to how coordinates change under a change of

bases (or coordinate system). Components of vectors transform contravariantly, whilecomponents of covectors (linear functionals) transform covariantly.

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The unitary basis can be described in terms of the rectangular Cartesian basis

as follows;

Cartesial

Rectangular basis

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Using the summation convention discussed in previous class;

* This should be a subscript

Summation

convention

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We can also construct another basis by taking the scalar product of vectorA with the

cross product ofe1xe2 and noting that since e1 e2 is perpendicular to both e1 and e2, we

obtain;

Remember!!

i=3

And solving for A3 we have;

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In similar fashion, we can obtain expressions for A1 and A2

e3

e2

e1

Therefore, the set of vectors (e1 , e2 , e3 ) is called the dual basis or reciprocal basis

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It is possible, since the dual basis is linearly independent to express a vectorA in terms

of the dual basis;

Notice now that the components associated with the dual basis have subscripts, andAi

are the covariant componentsofA.

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If we now returning to the scalar field , the differential change is given by;

Remember that these are components of dx

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We can now write d in such a way that we elucidate the direction as well as the

magnitude ofdx

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If we denote the magnitude of dx by ds=IdxI Then =dx/ds is a unit vector in the direction of dx

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The derivative (d/ds) is called the directional derivative of.

We see that it is the rate of change of with respect to distance and that it depends on

the direction in which the distance is taken.

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MAE 640 Lec4