Tensor Tutorial

SIMILAR workshop, June 2005

Tensor tutorial

SIMILAR workshop, 810 June 2005

Magnus HerberthsonDep. of Mathematics & Dep. of Biomedical Engineering

Linköping Universityemail: [email protected]


Tensor tutorial• What is a tensor?• What are tensors good for/examples?• Vector spaces, linear mappings• Change of basis• Dual vectors, dual space• The metric/scalar product, (& the determinant)• Tensors• Examples • Differentiation


What is a tensor?What is a vector?What is a linear mapping?

(Or rather, how doyou view these?)

Several opinions/viewpoints. My opinion: A mapping, a geometric object. Not a bunch of numbers.

N.B. notation tensors at a point ~ tensor fields vector space, affine space, point space


What are they good for?Tensors describe physical properties of/in mechanics (stress tensor, strain tensor) vector calculus (vector fields) medicin (wait and see...) electromagnetism (Maxwell tensor, EMfield) relativity (curvature of spacetime) ...


What is a vector?Example: a vector in the plane (with origin). Is it a, b or c?

(My choice is c.)


What about a)?

x

y

1

p

q

1

2

p

q

2


Vector spaces (over R)

uv

u+v

3/2 u


A picture

(What is this box for?)

• The elements are vectors• No basis: no coordinates• Beware: Rn is special

(why?)

N.B.vector space ~ affine space ~ point space


Notation, a remark


An example


Linear mappings

V W


The matrix of a mapping (given a basis)V W


Change of basis?

w2w1

v2

v1 x1v1

x2v2

y1w1

y2w2u


What happens to the coordinates of u?


What happens to the matrix ?


Gradient & scalar product as a mapping


The dual space

Ex. condensator:Potential V1

Potential V0Origin


How do we draw dual vectors?

Do you see that


An observation


The dual basis (cf. reciprocal basis)

(dual)

(reciprocal)


The dimension of V*


Coordinates, example


What happens to the coordinates of ω?


The metric, scalar productu v

θ


An observation


The components of g (what basis?)


The components of g (what basis?) II


Components of g, III


Another example


Another example II


But then....


Drawing the metric g

gu

v

g(u, ).


One definition of a tensor


Remember our list:• Tensors describe physical properties of/in• mechanics (stress tensor, strain tensor)• vector calculus (vector fields)• medicin (wait and see...)• electromagnetism (Maxwell tensor, EM

field)• relativity (curvature of spacetime)• ....


Some remarks• Have not talked so much of the basis • More can be said about notation• Tensors a point > tensor fields• Transformation properties of components

will follow• Differentiation of tensor fields, in particular

covariant derivative


A word on covariant derivative

A covector. Components.


Covariant derivative


Covariant derivative,II

That’sall fornow.

Tensor Tutorial

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