What is Vector calculus: Definition and 419 Discussions

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space





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{\displaystyle \mathbb {R} ^{3}.}
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 § Generalizations below for more).

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  1. C

    Good book about Vector Calculus?

    I'm studying Vector Calculus right now, and I'll have a test about Coordinate Transformation soon. But the book my teacher recommended (Mathematical Methods for Physicists - Arfken) is way too hard to understand this subject. Does anyone know any good material about this that I can find on...
  2. C

    Line Integral - Vector Calculus

    Homework Statement Calculate the line integral: f(x,y) = (x² - 2xy)î + (y² - 2xy)j, between the points (-1,1) and (1,1) along the parabola y = x². (resp: -14/15) The attempt at a solution I thought something like this: substitue y = x², and then integrate de f(x,y). And then evaluate...
  3. L

    Proving vector calculus identities using summation notation

    Homework Statement \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Equations \vec{r} = x_{i}e_{i} The Attempt at a Solution \frac{∂x_{i}}{∂x_{j}} = 1 iff i=j δ_{ij} = 1 iff i=j therefore \frac{∂x_{i}}{∂x_{j}} = δ_{ij} Homework Statement r^{2} = x_{k}x_{k} Homework...
  4. M

    Vector calculus, normals to plane curves

    Homework Statement show that n(t)=-g'(t)i+f'(t)j and -n(t)=g'(t)i-f'(t)j are both normal to the curve r(t)=f(t)i+g(t)j at the point (f(t),g(t)). Homework Equations The Attempt at a Solution i tried finding the unit normal of r(t) in hopes that it would be exactly what n(t) and...
  5. lonewolf219

    Vector calculus velocity question

    Hi, I am not fully understanding how to express velocity as a vector equation. In an example my professor gave in class, a particle moves (in a straight line) from point A to point B (starting from t=0) and traveling at 6 m/s. The solution included a unit vector. Does that mean all vector...
  6. C

    Proof - Vector Calculus - Curl

    I need to prove this: u x (\nabla x u) = \frac{1}{2}\nabla(u²) - (u \cdot \nabla)u. I've came to this: uj∂iuj - uj∂jui (i think it's correct) But how this 1/2 appears?
  7. Y

    Vector Calculus - Equations for planes tangent to given equation

    Homework Statement My problem is one pertaining to my Vector Calculus course. The assignment is asking us to "Find equations for the planes tangent to z = x2 + 6x + y3 that are parallel to the plane 4x − 12y + z = 7." The problem I'm having with the problem is the plural aspect. It states...
  8. H

    Short (I think) vector calculus question

    Homework Statement Calculate the following expressions: Homework Equations The Attempt at a Solution Letting the vector a = (a_1, a_2, a_3) I've worked out that it's 2|a|^2 While that method is fairly quick, I don't particularly like it, and was wondering if there is a shorter or neater...
  9. D

    Vector Calculus: Find Position & Velocity Vectors

    Homework Statement Use the given information to find the position and velocity vectors of the particle. a(t) = −cos t i − sin t j; v(0) = i; r(0) = j Homework Equations The Attempt at a Solution Ok first step integrate a(t). which i get to be -sin(t)i +cos(t)j + c now...
  10. D

    Vector calculus identities navigation

    Homework Statement I'm reading in a fluid dynamics book and in it the author shortens an equation using identities my rusty vector calculus brain cannot reproduce. Homework Equations \vec{e} \cdot \frac{\partial}{\partial t}(\rho \vec{u}) = -\nabla\cdot (\rho\vec{u})\cdot\vec{e} -...
  11. C

    Spivak's level vector calculus book

    Besides Apostol 2, is there any good, rigorous and suited for self study book on this subject? Thanks
  12. E

    Partial differentiation of cos (in vector calculus)

    Homework Statement So using standard spherical polar co-ordinates, my notes define a sphere as r(s,t) = aCos(s)Sin(t) i + aSin(s)Sin(t) j + aCos(t) k and the normal to the surface is given by the cross product of the two partial differentials: \partialr/\partials X \partialr/dt...
  13. A

    Angle between tangent and foci. (Involve vector calculus)

    Homework Statement See the photo on the left Homework Equations The Attempt at a Solution
  14. S

    Vector Calculus Supplies: Textbook, Syllabus, & More

    Looking for supplemental material. This is the textbook I am supposed to use. (http://www.amazon.com/dp/0321549287/?tag=pfamazon01-20) The syllabus is: Vector fields; vector calculus; ordinary differential equations; sequences, series, and power series.
  15. L

    Curvilinear Coordinates and Vector Calculus

    Homework Statement With \vec{L} = -i\vec{r} x \nabla, verify the operator identities \nabla = \hat{r}\frac{\partial }{\partial \vec{r}}-i\frac{\vec{r}\times\vec{L}}{r^{2}} and \vec{r} \bigtriangledown ^2 - \bigtriangledown (1+\vec{r}\frac{\partial }{\partial \vec{r}})=i\bigtriangledown \times...
  16. Z

    Apostle, chapter 14 problems: Vector Calculus

    Homework Statement Problem 1: Two fixed unit vectors A and B make an angle θ with each other, where 0 < θ < π. A particle moves in a space curve in such a way that its position vector r(t) and velocity v(t) are related by the equation v(t) = A x r(t). If r(0) = B, prove that the curve has...
  17. R

    Vector calculus identities and maxwell equations

    so we have the identity \nabla\times\nabla\phi = 0 and from Maxwell's equations we have \nabla\times \textbf{E} = -\frac{d\textbf{B}}{dt} But we also have that \textbf{E} = -\nabla\phi So the problem I'm having is this -\textbf{E} = \nabla\phi which i substitute into the...
  18. F

    Understanding vector calculus proofs

    ive been trying to understand a few of the identities my professor gave me and i can get a few of them down such as \nabla(\vec{A}\vec{B})=\vec{B}\nabla\vec{A} - \vec{A}\nabla\vec{B} and i can break it down through cartesian and product rules but when i try to do \nabla X (\vec{A}ψ) =...
  19. R

    Proving the Vector Calculus Identity: (1/g^2)(g∇f - f∇g)

    I am trying to figure out a proof for this identity \nabla(f/g) = (1/g2) (g\nablaf - f\nablag) Any ideas?
  20. G

    Useful Tricks for Calculus & Vector Calculus

    Can anyone list some useful tricks for calculus and vector calculus. i.e. sneaky substitutions, changing limits of integration when changing coordinate systems, parametrization of curves, surfaces, and volumes etc.
  21. M

    Gradient, unit normal in vector calculus

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  22. S

    Vector calculus book recommendations

    Hi, i have just finished self-studying spivak calculus and have thoroughtly enjoyed reading it and doing the problems. I am looking to find a book on vector calculus with similar rigor as that of spivak. Any recommendations? I have heard Hubbards book on vector calculus is not bad. Any...
  23. G

    Find the Perfect Vector Calculus Book: Expert Recommendations

    Hi! Could you recommend a book related with vector calculus? I want to get a book which is written so rigorously that it helps me understand the topic 'vector calculus' more precisely. I'll wait for your good recommendations. Thanks!
  24. I

    Two problems involving vector calculus-

    Homework Statement (Sorry for the confusing font- I tried to figure out using the latex reference to indicate a vector, but couldn't do it- please view the "superscripted" variable as a vector. Thanks) A. The electrostatic potential at P arising form a dipole of unit strength at O...
  25. T

    Vector calculus and force field

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  26. A

    Vector calculus question regarding helmholtz theorem

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  27. C

    Proving vector calculus identities using the levi-civitia symbol

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  28. P

    Proving Vector Calculus Identity Without Introducing Coordinates

    Can someone help me prove the identity \ u \times (\nabla \times u) = \nabla(u^2 /2) - (u.\nabla)u without having to write it out in components?
  29. 6

    Vector calculus evaluation question

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  30. H

    Vector Calculus & Diff Equations book?

    Ok, the professor I got who is supposed to teach Vector Calculus and Differential Equations sucks and also does the book he uses. I am a Mechanical Engineering student and I will begin self teaching those subjects. For that matter, I would like good books, if possible. I sense those subjects are...
  31. S

    Proving Vector Calculus: Cyclic Integral of (r dot ds)=0

    How to prove that cyclic integral of (r dot ds) =0 (symbols having usual meanings). Please help me 2) It is always possible to find curl when vector function is known, but how to find the vector when its curl is known.
  32. Goddar

    Vector calculus eq. needs translation

    Homework Statement Hi, this is not part of a problem but just an equation I'm having a hard time to decipher (for the reference the original one is in "Statistical Mechanics" by Pathria, eq. 3.7.16) We define: r = |r2–r1|, Where bold letters are vectors, and we basically integrate a function...
  33. Q

    Vector calculus identities - is this right?

    I'm manipulating an equation, and I think I am correct in doing this, but not sure. Could someone tell me if the equality I've written below is true? [\nabla\cdot [\rho\vec{v}\vec{v}] ]\cdot\vec{v} = \nabla\cdot[\frac{1}{2}\rho v^2 \vec{v}] (where \rho is dependent on position) *NOTE* that...
  34. Q

    A fairly easy vector calculus identity question?

    I'm working on simplifying a big physical expression (I don't like the Navier-Stokes equations at all anymore), and I'm curious how to simplify the following term: \vec{v}\cdot (\vec{v}\cdot\nabla )\vec{v} where v is a fluid velocity - i.e. definitely spatially varying. I'm just not sure...
  35. N

    Dirac delta, generalizations of vector calculus and sigh vagueness

    Although I am an aspiring physicist, I cannot cope with the physicist's love for vagueness when it comes to yielding math. Exactness is simply not a luxury that can be ignored, certainly not in theoretical physics. But okay, I realize the dirac delta function can be made exact by the use of...
  36. T

    Vector Calculus Homework: Solving Part (a) & (b)

    Homework Statement [PLAIN]http://img576.imageshack.us/img576/1710/vectorp.png Homework Equations The Attempt at a Solution I've done part (a) but how do I do (b)?
  37. T

    Vector Calculus: Index Notation

    Homework Statement [PLAIN]http://img585.imageshack.us/img585/526/indexnotation.jpg The Attempt at a Solution How do I proceed?
  38. V

    Vector Calculus: Worth Dual Enrolling in High School?

    I'm in high school, and right now I'm taking AP Calculus. I'm interested in dual enrolling at a Community College over the summer. I've been looking at their Math selection, and they list Vector Calculus as a course. It sounds, from reading the...
  39. B

    How do you evaluate the vector calculus question over an ellipsoid?

    The question is Let r=r(x,y,z) where it is the distance from a point O. Evaluate \oint\nabla(1/r)*ndS (where * is the dot product) over the ellipsoid x^2/4 +y^2/9+z^2/25=1 I thought the answer was 0 since the ellipsoid is a simply connected region in R^3 and the...
  40. U

    Vector Calculus: Integral Theorems

    Homework Statement Question 3 part b and c Homework Equations Divergence and Stokes Theorems. Knowledge of parametrization ect ect The Attempt at a Solution I got the B field by using curl. However any attempt to resolve the flux through the top hemisphere or even the...
  41. P

    Vector Calculus: Dots, Crosses & Triple Products Explained

    Can anybody please suggest me a good book which covers following topics in detail Vectors - Dots, Cross and triple products, Gradient, divergence and applications. thnx in advance
  42. S

    Proof of Conservation of Vector Calculus: Force on a Mass with Position Vector r

    Homework Statement The Force on a mass with position vector r satisfies: m\frac{d^{2}\textbf{r}}{dt^{2}}=F=f(\textbf{r})\textbf{r} where f(r) is scalar function of r. Show that L: L=\textbf{r}\times\frac{d\textbf{r}}{dt} is conserved. Homework Equations The Attempt at a...
  43. A

    Proving Harmonicity of g in Vector Calculus | Closed Ball & Surface Sketch

    Homework Statement Let g: D-->R. D subset of R^3 be harmonic Then for any closed ball that is a subset of D with radius >0 and its origin in D With its surface s=(partial d)B(a,r) = {x={x1,x2,x3} st mod(x-a) = r} show that f(a)=(1/4.pi.r^2)INTRGL: f (over s) Homework Equations...
  44. E

    About exterior algebra in vector calculus

    I'm reading Marsden's vector calculus. In the chapter of differential forms, it mentions the wedge product satisfies the laws: dy^dx=-dxdy. and for a 0-form f, f^w=fw. Does it have formal derivation? hope someone can give me a hint or even a link.
  45. M

    Solve for Area with Given Vertices Using Vector Calculus

    Homework Statement find the area if the vertices are (3,9,8),(0,5,1),(-1,-3,-3),(2,1,4) Homework Equations The Attempt at a Solution I draw the points and I couldn't know the shape it is complex I really couldn't know it
  46. C

    Vector Calculus: Computing (V•∇)U and (U•∇)V with Given Functions

    Homework Statement Hello. I want to see if I am interpreting the following correctly, I certainly don't expect anyone to work the problem out as it is (at least with my approach) fairly tedious. Compute the following: (\vec{V}\cdot\nabla)\vec{U} (\vec{U}\cdot\nabla)\vec{V} Given: \vec{r} =...
  47. F

    Book recommendation for vector calculus.

    Hello, I'm learning vector calculus in my physics class. We're using Mathematical methods for Physicists by Arfken, but I think it makes a better reference source than something to learn the concepts from. I've downloaded the Feynman lectures which seem to be pretty good so far, but I was...
  48. P

    Vector calculus and order of operations

    I understand how to do (a). However, I'm having trouble with the rest, here. I've never properly learned the order of operation for vectors like this. For example, for (b) does one dot (r - r') with itself BEFORE using the del operator? Thanks guys, help appreciated. Even if you point me...
  49. D

    Vector Calculus Question (I don't understand)

    Homework Statement Let D* = [0,1]x[0,1] and define T on D* by T(u,v)=(-u^2+4u, v). Find the image D. Is T one-to one Homework Equations The Attempt at a Solution I have no idea... I don't know how to do it. The solution is [0,3] x [0,1]... yes it is one to one. am I supposed...
  50. D

    Vector calculus and flight path

    A pilot flies with a heading of 160 degrees and an airspeed of 250km/h. a)how long should it take the pilot to fly to a town that is 1200km away on the heading he has chosen b) there is a steady wind of 30km/h from the drection 030 degrees. Calculate the ground velocity c) How far, and...
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