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SIMILAR workshop, June 2005
Tensor tutorial
SIMILAR workshop, 810 June 2005
Magnus HerberthsonDep. of Mathematics & Dep. of Biomedical Engineering
Linköping Universityemail: [email protected]
SIMILAR workshop, June 2005
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
SIMILAR workshop, June 2005
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
SIMILAR workshop, June 2005
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) ...
SIMILAR workshop, June 2005
What is a vector?Example: a vector in the plane (with origin). Is it a, b or c?
(My choice is c.)
SIMILAR workshop, June 2005
What about a)?
x
y
1
p
q
1
2
p
q
2
SIMILAR workshop, June 2005
Vector spaces (over R)
uv
u+v
3/2 u
SIMILAR workshop, June 2005
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
SIMILAR workshop, June 2005
Notation, a remark
SIMILAR workshop, June 2005
An example
SIMILAR workshop, June 2005
Linear mappings
V W
SIMILAR workshop, June 2005
The matrix of a mapping (given a basis)V W
SIMILAR workshop, June 2005
Change of basis?
w2w1
v2
v1 x1v1
x2v2
y1w1
y2w2u
SIMILAR workshop, June 2005
What happens to the coordinates of u?
SIMILAR workshop, June 2005
What happens to the matrix ?
SIMILAR workshop, June 2005
Gradient & scalar product as a mapping
SIMILAR workshop, June 2005
The dual space
Ex. condensator:Potential V1
Potential V0Origin
SIMILAR workshop, June 2005
How do we draw dual vectors?
Do you see that
SIMILAR workshop, June 2005
An observation
SIMILAR workshop, June 2005
The dual basis (cf. reciprocal basis)
(dual)
(reciprocal)
SIMILAR workshop, June 2005
The dimension of V*
SIMILAR workshop, June 2005
Coordinates, example
SIMILAR workshop, June 2005
What happens to the coordinates of ω?
SIMILAR workshop, June 2005
The metric, scalar productu v
θ
SIMILAR workshop, June 2005
An observation
SIMILAR workshop, June 2005
The components of g (what basis?)
SIMILAR workshop, June 2005
The components of g (what basis?) II
SIMILAR workshop, June 2005
Components of g, III
SIMILAR workshop, June 2005
Another example
SIMILAR workshop, June 2005
Another example II
SIMILAR workshop, June 2005
But then....
SIMILAR workshop, June 2005
Drawing the metric g
gu
v
g(u, ).
SIMILAR workshop, June 2005
One definition of a tensor
SIMILAR workshop, June 2005
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)• ....
SIMILAR workshop, June 2005
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
SIMILAR workshop, June 2005
A word on covariant derivative
A covector. Components.
SIMILAR workshop, June 2005
Covariant derivative
SIMILAR workshop, June 2005
Covariant derivative,II
That’sall fornow.