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Faculty of Mechanical Engineering Engineering Computing Panel

MKMM 1213 Advanced Engineering Mathematics Covariant & Contravariant Vector Components

Abu Hasan Abdullah

May 2015

abu.hasan.abdullahdev.null

MKMM 1213 Advanced Engineering Mathematics

Covariance & Contravariance

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Outline I

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Covariant & Contravariant Vector Components Terminologies Covariant Transformation Contravariant Transformation Change of Basis Rotation of the Coordinate Axes

2

Bibliography




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Covariant & Contravariant Vector Components Terminologies

Transformation in a vector space is a rule that specifies how certain entities change under a change of basis. A basis for a vector space of dimension n is a sequence of n vectors, (e1 , . . . en ), with the property that every vector in the space can be expressed uniquely as a linear combination of the basis vectors. Covariant and contravariant are terms much used in the transformation vectors and tensors. A covariant transformation describes the new basis vectors as a linear combination of the old basis vectors. A contravariant transformation is the inverse of a covariant transformation.




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Covariant & Contravariant Vector Components Covariant Transformation

In Figure 1 a vector v is described in a Cartesian coordinate grid (black lines) whose basis vectors are ex and ey . In another (cylindrical polar) coordinate system (dashed red lines), the new basis vectors are tangent vectors in the radial direction, er and perpendicular to it, eϕ , which appear rotated anticlockwise with respect to the first basis. Figure 1: A vector v, and local tangent basis vectors ex , ey and er , eϕ .

The covariant transformation here is thus an anticlockwise rotation.




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Covariant & Contravariant Vector Components Contravariant Transformation

If we view the vector v with eϕ pointed upwards, its representation in this frame appears rotated to the right. The contravariant transformation is a clockwise rotation. Figure 2: Representations of v in two coordinate systems.

The vector itself is a geometrical quantity, in principle, independent (invariant) of the chosen basis. A vector v is given, say, in components vi on a chosen basis ei . On another basis, say e′j , the same vector v has different components v′j and X ′j ′ X i v ej v ei = v= i

j

By convention, lower indices identify the basis vectors upper indices identify the components of a vector abu.hasan.abdullahdev.null



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Covariant & Contravariant Vector Components Change of Basis

Let V be a vector space and let S = (v1 , v2 , . . . , vn ) be a set of vectors in V. Recall that S forms a basis for V if the following two conditions hold: S is linearly independent S spans V

If S = (v1 , v2 , . . . , vn ) is a basis for V, then every vector v ∈ V can be expressed uniquely as a linear combination of v1 , v2 , . . . , vn : v = c1 v1 + c2 v2 + . . . + cn vn where [c1 c2 . . . cn ]T are coordinates of v relative to the basis S. If V has dimension n, then every set of n linearly independent vectors in V forms a basis for V. In every application, there’s a choice as to what basis to use. Will describe the transformation of coordinates of vectors under a change of basis, focus on vectors in R2 , although all of this generalizes to Rn .




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The standard basis in R2 is {[ 10 ] , [ 01 ]}. We specify other bases with reference to this rectangular coordinate system. From Figure 1, let B = {ex , ey }

and

B′ = {er , eϕ }

be two bases for R2 . For a vector v ∈ V , given its coordinates [v]B in basis B we would like to be able to express v in terms of its coordinates [v]B′ in basis B′ , and vice versa. Suppose the basis vectors er and eϕ for B′ have the following coordinates relative to the basis B: » – a = aex + bey [er ]B = b » – c = cex + dey [eϕ ]B = d




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The “change-of‘coordinates” matrix from B′ to B – » a c P= b d governs the change of coordinates of v ∈ V under the change of basis from B′ to B. » – a c [v]B′ [v]B = P[v]B′ = b d That is, if we know the coordinates of v relative to the basis B′ , multiplying this vector by the change of coordinates matrix gives us the coordinates of v relative to the basis B. The transition matrix P is invertible. If P is the change of coordinates matrix from B′ to B, then P−1 is the change of coordinates matrix from B to B′ : [v]B′ = P−1 [v]B




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1 0 Example ˘ 1:ˆ Let˜¯B = {[ 0 ] , [ 1 ]} and B′ = [ 31 ] , −2 . The change of basis 1 matrix from B′ to B is – » 3 −2 . P= 1 1

The vector v with coordinates [v]B′ = [ 21 ] relative to the basis B′ has coordinates –» – » – » 4 3 −2 2 = [v]B = P[v]B′ = 3 1 1 1 relative to the basis B. Since » 1 – 2 P−1 = 5 1 53 , −5 5 we can verify that » 1 [v]B′ = P−1 [v]B = 5 1 −5 abu.hasan.abdullahdev.null

Figure 3: Change of two bases.

2 5 3 5

–» – » – 2 4 = 1 3



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Example 2: In this example, we introduce a third basis, B′′ to look at the relationship between two non-standard bases, i.e. B′ and B′′ . ˘ ˆ ˜¯ Let B′′ = [ 21 ] , 14 . Find the change of coordinates matrix from the basis B′ of Example 1 to B′′ , and express the vector v with coordinates [ 21 ] relative to the basis B′ . Figure 4: Change of three bases.




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Covariant & Contravariant Vector Components Rotation of the Coordinate Axes

Suppose we obtain a new coordinate system from the standard rectangular coordinate system, see Figure 5, by rotating the axes counterclockwise by an angle θ. The new basis B′ = {u′ , v′ } of unit vectors along the x′ - and y′ -axes, respectively, has coordinates – – » » − sin θ cos θ ′ ′ , [w ]B = [u ]B = cos θ sin θ in the original B coordinate system.

Figure 5: Rotation of axes.

Thus » cos θ P= sin θ


− sin θ cos θ

–

and

−1

P

»

cos θ = − sin θ

sin θ cos θ


–


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Covariant & Contravariant Vector Components Rotation of the Coordinate Axes

The vector [ xy ]B in the original coordinate system has coordinates » » ′– cos θ x = ′ − sin θ y B′

sin θ cos θ

h ′i x y′

B′

given by

–» – x y B

in the rotated coordinate system.




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Bibliography 1

P ETER V. O’N EIL (2012): Advanced Engineering Mathematics, 7ed, ISBN-13: 978-1-111-42741-2, Cengage Learning

2

D ANIELA F LEISCH (2012): A Student’s Guide to Vectors and Tensors, ISBN: 978-0-521-17190-8, Cambridge University Press

3

E RWIN K REYSZIG (2011): Advanced Engineering Mathematics, 10ed, ISBN: 978-0-470-45836-5, John Wiley & Sons

4

A LAN J EFFREY (2002): Advanced Engineering Mathematics, ISBN: 0-12-382592-X, Harcourt/Academic Press

5

G LYNN JAMES ET AL . (2011): Advanced Modern Engineering Mathematics, 4ed, ISBN: 978-0-273-71923-6, Pearson Education

6

L. B RIGGS ET AL . (2013): Calculus for Scientists and Engineers: Early Transcendentals, ISBN-13: 978-0-321-78537-4, Pearson Education




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