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Continuum mechanics MAE 640
Summer II 2009
Dr. Konstantinos Sierros
263 ESB new add
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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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Vector calculus Derivative of a scalar function of a vector
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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Vector calculus Derivative of a scalar function of a vector
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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Vector calculus Derivative of a scalar function of a vector
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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Vector calculus Derivative of a scalar function of a vector
The unitary basis can be described in terms of the rectangular Cartesian basis
as follows;
Cartesial
Rectangular basis
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Vector calculus Derivative of a scalar function of a vector
Using the summation convention discussed in previous class;
* This should be a subscript
Summation
convention
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Vector calculus Derivative of a scalar function of a vector
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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Vector calculus Derivative of a scalar function of a vector
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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Vector calculus Derivative of a scalar function of a vector
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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Vector calculus Derivative of a scalar function of a vector
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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Vector calculus Derivative of a scalar function of a vector
We can now write d in such a way that we elucidate the direction as well as the
magnitude ofdx
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Vector calculus Derivative of a scalar function of a vector
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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Vector calculus Derivative of a scalar function of a vector
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.