Tensor analysis

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors". Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing the earlier work of Bernhard Riemann, Elwin Bruno Christoffel, and others – as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor. Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction. A tensor may be represented as an (potentially multidimensional) array.

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Related exercises (32)

Elliptical trajectory

A material point of mass m is moving in the plane defined by the orthonormal coordinate system Oxy such that its position vector is given by r⃗u2004=u2004Acosu2006(ωt)u2006îu2005+u2005Bsinu2006(ωt)u2006ĵ where A, B and ω are positive constants and î and ĵ are the unitary vectors of axes Ox and Oy.

Show that the material point’s path is elliptical. Sketch the velocity v⃗ and acceleration a⃗ vectors along the trajectory. Show that if Au2004≠u2004B, then vectors r⃗(t) and v⃗(t) are, generally,

Rotation on a two-blade helix

A homogeneous rod of length L and mass M rotates in a vertical plane around the horizontal axis OG at constant angular velocity ψ̇u2004=u2004ψ̇u2006e₃. The axis OG rotates around the vertical axis ẑ at constant angular velocity ϕ̇u2004=u2004ϕ̇u2006e_(z). The moment of inertia of the rod with respect to its axis is I_(G∥) and the torques perpendicular to the axis of the rod passing through G are I_(G⊥). The distance |OG|u2004=u2004R.

Show that the angular momentum L_(G) is expressed in the principal

Conceptual questions on cross product

Let O be the centre of a watch. We define an x-axis along the minute hand and a z-axis along the hour hand. In which direction does the y-axis point, knowing that it forms a direct orthonormal coordinate system Oxyz with the other two, when the time is 9xa0am? How about when it’s 3xa0pm?
Knowing that the sun rises East and sets West, is the vector for the Earth’s angular velocity oriented from the North Pole to the South Pole, or from the South Pole to the North Pole?
To

Ellipse

A material point moves in the vertical plane Oxy such that its position vector is given by ru2004=u2004acosu2006(ωt)e_(x)u2005+u2005bsinu2006(ωt)e_(y), where a and b are positive constants such that au2004>u2004b, and e_(x) and e_(y) are the unit vectors of the axis Ox and Oy.

Show that the material point moves on an ellipse.

A material point M moves in a plane containing the fixed points A and B such that the condition ∥AM∥u2005+u2005∥MB∥u2004=u2004const is verified.

Show that the material poin

Wheel on a tilted axis

A wheel with radius R is attached, in its centre A to the end of a rigid axis of length d, perpendicular to the wheel’s plane. The other end of the axis is secured to a point O on the ground, which is assumed to be flat (see diagram). The wheel rolls, without sliding, on the ground, because the axis has a rotational motion characterised by an angular velocity vector Ω⃗(t) pointing vertically upwards.

Using the formula linking the respective velocities of all points of a non-deformable solid, find t

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