Vector analysis

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra which uses exterior products does (see below for more). Scalar field A scalar field associates a scalar value to every point in a space. The scalar is a mathematical number representing a physical quantity. Examples of scalar fields in applications include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields (known as scalar bosons), such as the Higgs field. These fields are the subject of scalar field theory. Vector field A vector field is an assignment of a vector to each point in a space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. This can be used, for example, to calculate work done over a line.

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Related exercises (32)

Irregular tube

An incompressible liquid circulates in a tube. At a certain location, the section of the tube strongly decreases (cf. figure). The continuity equation indicates us that :

\frac{\partial\rho}{\partial t} + \nabla\cdot \rho \vec{v} = 0~.

Show that this is equivalent to the simpler equation:

\quad \langle v_n\rangle=\frac{1}{S_n}\iint_{S_n}\vec{v}\cdot d\vec{\sigma} ~.$$ If we add to the figure the velocity vectors as well as the vect

Hole-in-the-table

A horizontal table has a hole that allows a thread of length L to pass without friction. This thread connects two blocks: 1) a block of mass m that slides without friction on the table and whose initial velocity is not in the thread’s direction and 2) a block of mass M that’s vertically hanging below the table. The thread is always taut.

Write the blocks’ total mechanical energy, considering that the thread has constant length.
What are the conserved quantities, that is, the primary integrals

Moment of inertia of a sphere

xa0 Calculate the moment of inertia I_(z) of a sphere that is full and homogeneous with a mass M and with a radius R, around an axis (Oz) going through its centre of mass.

Make this calculation while noticing that by symmetry, for the three perpendicular axes (Ox), (Oy) et (Oz) we have I_(x)u2004=u2004I_(y)u2004=u2004I_(z).
Optional Redo this but starting from a volume element in spherical coordinates. Formulary : the primitive of sin³θ is $(\frac{1}{3}\cos^3\theta - \cos\theta)$ .

Vortex of Rankine

The vortex of Rankine is a model of a fluid defined by the velocity field (Ω=const):

{\bf v} = \Omega \,r \,{\bf\hat{\phi}} \quad r\leq a~; \qquad {\bf v} = \frac{\Omega\, a^2}{r} \,{\bf\hat{\phi}} \quad r> a~.

The flow is stationary, the fluid uncompressible, and we will neglect gravity. Determine the vorticity

({\bf T}=\nabla \times {\bf v})

of the flow as well as the pressure field P(r) with the condition P(r→∞)u2004=u2004P_(∞). Suggestion: use the vectorial identity: $({\bf v}\cdot \nabla){\bf v

Related notebooks (16)

Illustrating Band Structures in Solids: Free Electrons and Atomic Orbitals

In [1]: import numpy as np from numpy import linalg as LA import matplotlib.pyplot as plt

from future import print_function

from ipywidgets import interact, interactive, fixed, interact_manual import ipywidgets as widgets %matplotlib widget The emergence of band structures[1]¶ If you have questions about this notebook, e-mail Oleg Malanyuk (oleg.malanyuk@epfl.ch) This jupyter notebook is intended to provide a illustration of the band structures in solides, starting from the points of v

Tight-Binding Approximation for 2D Lattices: Square, Triangular, Lieb, and Honeycomb

In [1]: import numpy as np import matplotlib.pyplot as plt

from future import print_function

from ipywidgets import interact, interactive, fixed, interact_manual import ipywidgets as widgets %matplotlib inline In [2]: def band(nkl,kpoints,ham): :param k: wave number nk=32 # Define the position of the original lattice #klines=np.zeros((nkl-1,nk)) n=0 d=np.size(ham([0.0,0.0]),1) evk=np.zeros((nknkl-nk-1,d)) lenk=np.zeros((nknkl-nk-1,d)) for i in range(0,nkl-1)

Understanding the Haar Basis for Grayscale Image Compression

A Basis for Grayscale Images[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

The concept of "basis" for a vector space is one of the most profound and fruitful ideas in linear algebra. Basis vectors are the "building blocks" that all other vectors are built from; by "disassembling" a vector into a linear combination of basis elements we are able to "look inside" the vector and highlight many of its relevant features. A vector space has an infinite number o

Modeling Epidemics with Signal Processing Techniques

Related exercises (32)

Irregular tube

An incompressible liquid circulates in a tube. At a certain location, the section of the tube strongly decreases (cf. figure). The continuity equation indicates us that :

\frac{\partial\rho}{\partial t} + \nabla\cdot \rho \vec{v} = 0~.

Show that this is equivalent to the simpler equation:

\quad \langle v_n\rangle=\frac{1}{S_n}\iint_{S_n}\vec{v}\cdot d\vec{\sigma} ~.$$ If we add to the figure the velocity vectors as well as the vect

Hole-in-the-table

Write the blocks’ total mechanical energy, considering that the thread has constant length.
What are the conserved quantities, that is, the primary integrals

Moment of inertia of a sphere

xa0 Calculate the moment of inertia I_(z) of a sphere that is full and homogeneous with a mass M and with a radius R, around an axis (Oz) going through its centre of mass.

Make this calculation while noticing that by symmetry, for the three perpendicular axes (Ox), (Oy) et (Oz) we have I_(x)u2004=u2004I_(y)u2004=u2004I_(z).
Optional Redo this but starting from a volume element in spherical coordinates. Formulary : the primitive of sin³θ is $(\frac{1}{3}\cos^3\theta - \cos\theta)$ .

Vortex of Rankine

The vortex of Rankine is a model of a fluid defined by the velocity field (Ω=const):

{\bf v} = \Omega \,r \,{\bf\hat{\phi}} \quad r\leq a~; \qquad {\bf v} = \frac{\Omega\, a^2}{r} \,{\bf\hat{\phi}} \quad r> a~.

The flow is stationary, the fluid uncompressible, and we will neglect gravity. Determine the vorticity

({\bf T}=\nabla \times {\bf v})

of the flow as well as the pressure field P(r) with the condition P(r→∞)u2004=u2004P_(∞). Suggestion: use the vectorial identity: $({\bf v}\cdot \nabla){\bf v

Epidemics and Signal Processing[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In [1]: import matplotlib.pyplot as plt import numpy as np import scipy.signal as sp import seaborn as sns sns.set() In [2]: plt.rcParams["figure.figsize"] = (14,4) The basic reproduction number[4]¶ The propagation mechanism of an epidemic such as the one caused by SARS-CoV-2 virus can be roughly modeled, at least in its inital phase, as a process in which each infected individ

Vector analysis

Chat with Graph Search

from future import print_function

from future import print_function

from future import print_function

from future import print_function

Adaptive vector-valued splines for the resolution of inverse problems

Sharp nonuniqueness in the transport equation with Sobolev velocity field

Mimetic finite difference schemes for transport operators with divergence-free advective field and applications to plasma physics