Vector calculus Definition and 422 Threads

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. T

    Calculus Vector Calculus Hubbard enough for Physics?

    Hello everyone, How far can the book take me before I have to read another book on the subject? Thanks :)
  2. J

    Index Notation Help: Solve [a,b,c]^2

    1. The problem is: ( a x b )⋅[( b x c ) x ( c x a )] = [a,b,c]^2 = [ a⋅( b x c )]^2 I am supposed to solve this using index notation... and I am having some problems. 2. Homework Equations : I guess I just don't understand the finer points of index notation. Every time I think I am getting...
  3. B

    Calculus Theoretical Multivariable Calculus books

    Dear Physics Forum advisers, Could you recommend books that treat the multivariable calculus from a theoretical aspect (and applications too, if possible)? I have been reading Rudin's PMA and Apostol's Mathematical Analysis, but their treatment of vector calculus is very confusing and not...
  4. M

    Simplify the proof of different vector calculus identities

    Is there a way to simplify the proof of different vecot calculus identities, such as grad of f*g, which is expandable. And also curl of the curl of a field. Is there a more convenient way to go about proving these relations than to go through the long calculations of actually performing the curl...
  5. T

    Divergence of radial unit vector field

    Sorry if this was addressed in another thread, but I couldn't find a discussion of it in a preliminary search. If it is discussed elsewhere, I'll appreciate being directed to it. Okay, well here's my question. If I take the divergence of the unit radial vector field, I get the result: \vec...
  6. kostoglotov

    2 cylinders intersect, area of resulting parametric surface

    Homework Statement I want to know if I got the answer correct and if my reasoning is sound. The text answers and solutions manual only gives answers/solutions for odd numbered problems. Here is the problem: And a direct link to the imgur page: http://i.imgur.com/Tko1xFh.png Homework...
  7. D

    Coordinate charts and change of basis

    So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let M be an n-dimensional manifold and let (U,\phi) and (V,\psi) be two overlapping coordinate charts (i.e. U\cap V\neq\emptyset), with U,V\subset M, covering a neighbourhood of p\in M, such that...
  8. B

    Vector Calculus Swimming Problem

    Homework Statement [/B] A swimmer located at point A needs to reach a point B 20 meters downstream on the opposite bank of a 10 meter wide river. The river flows horizontally at a rate of 0.5 meters/second, and the swimmer has a constant speed of 0.25 meters/second. Set up the vector...
  9. D

    Differential map between tangent spaces

    I've been struggling since starting to study differential geometry to justify the definition of a one-form as a differential of a function and how this is equal to a tangent vector acting on this function, i.e. given f:M\rightarrow\mathbb{R} we can define the differential map...
  10. A

    Vector calculus - How to use the gradient?

    I have done part A (i think) really not sure where to begin with the rest of the parts, would appreciate a tip in the right direction, its revision for my first year physics exams in a few weeks. Consider the funtion T in the plane (x,y), given by T=ln(x^2 + y^2) at point 1,2 a) in which...
  11. kelvin490

    Question about conditions for conservative field

    Question about conditions for conservative field In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is simply connected and open. Usually in textbooks there is not much explanations on why these...
  12. D

    How to compute the surface height based on normal vectors

    Suppose I have already found the surface normal vectors to a set of points (x,y), how do I compute the surface height z(x,y)? Basically what I have are the normal vectors at each point (x,y) on a square grid. Then I calculate the vectors u = (x+1,y,z(x+1,y)) - (x,y,z(x,y)) and v =...
  13. L

    Can I pull a time derivative outside of a curl?

    Homework Statement For the equation ∇ x E = -∂B/∂t I took the curl of both sides to get ∇ x (∇ x E) = ∇ x -∂B/∂t I feel like it'd be very wrong to pull out the time derivative. Am I correct?
  14. inversquare

    What Comes After Vector Calculus in Pure Calculus?

    Are there fields of "pure" Calculus that follow Vector Calculus? I mean fields that are not primarily ODE, PDE etc, Real Analysis or Complex Analysis, topology etc. Does Vector Calculus and its prerequisites fully encompass the basics of Calculus as a field?
  15. V

    Divergence Theorem Question (Gauss' Law?)

    If F(x,y,z) is continuous and for all (x,y,z), show that R3 dot F dV = 0 I have been working on this problem all day, and I'm honestly not sure how to proceed. The hint given on this problem is, "Take Br to be a ball of radius r centered at the origin, apply divergence theorem, and let the...
  16. Robsta

    Construct B field from a given E field using Maxwell's Eqns

    Homework Statement Given an electric field in a vacuum: E(t,r) = (E0/c) (0 , 0 , y/t2) use Maxwell's equations to determine B(t,r) which satisfies the boundary condition B -> 0 as t -> ∞ Homework Equations The problem is in a vacuum so in the conventional notation J = 0 and ρ = 0 (current...
  17. B

    Calculus Regarding to Multi-Variable Calculus Books

    Dear Physics Forum personnel, I am a college sophomore with double majors in mathematics & microbiology and an aspiring analytic number theorist. I will be going to self-study the vector calculus by using Hubbard/Hubbard as a main text and Serge Lang as a supplement to Hubbard; this will help...
  18. S

    Intuitive interpretation of some vector-dif-calc identities

    Dear All, I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to...
  19. bananabandana

    Calculating Flux through Ellipsoid

    Homework Statement Let ## E ## be the ellipsoid: $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+z^{2}=1 $$ Let ## S ## be the part of the surface of ## E ## defined by: $$ 0 \leq x \leq 1, \ 0 \leq y \leq 1, \ z > 0 $$ Let F be the vector field defined by $$ F=(-y,x,0)$$ A) Explain why ##...
  20. C

    Visualizing a Parametric Equation in 3D Space

    Homework Statement Given the eqn x=2, y=sin(t), z=cos(t), draw this function in 3-space. Homework Equations ABOVE^ The Attempt at a Solution I did this: x^2+y^2+z^2=2^2+(sin(t))^2+(cos(t))^2=5 Therefore we get x^2+y^2+z^2=5 Which is the eqn of a sphere with radius root5. My friend said it's...
  21. D

    Equation of the tangent plane in R^4

    Let f: \mathbb R^2 \to \mathbb R^2 given by f=(sin(x-y),cos(x+y)) : find the equation of the tangent plane to the graph of the function in \mathbb R^4 at (\frac{\pi}{4}, \frac{\pi}{4}, 0 ,0 ) and then find a parametric representation of the equation of the tangent plane What I did: the...
  22. P

    Vector Calculus to derive Newton's second law

    THIS THREAD HAS BEEN MOVED FROM ANOTHER FORUM BECAUSE IT IS HOMEWORK. BUT THERE IS NO TEMPLATE. Hi everyone, The problem gives that a particle moves in a circle with angular velocity ω. I know that r×ω=v, which is the velocity of the particle. However, I am told to differentiate and find τ=Iα...
  23. A

    Vector Calculus: Air flowing through loop of straight lines

    Homework Statement Air is flowing with a speed v in the direction (-1, -1, 1,) calculate the volume of air flowing through the loop consisting of straight lines joining (in order i presume) (1,1,0) (1,0,0) (0,0,0) (0,1,1) (1,1,0) Homework EquationsThe Attempt at a Solution I assume you have to...
  24. C

    Vector calculus, + finding parametric equation

    Homework Statement The line L L1: x=3+2t, y=2t, z=t Intersects the plane x+3y-z=-4 at a point P. Find a set of parametric equations for the line in the same plane that goes through P and is perpendicular to L. Homework Equations cross-product r=r0+t(vector) this is to get in parametric form...
  25. JonnyMaddox

    Translation from vector calc. notation to index notation

    Hi, I want to translate this equation R_{\hat{n}}(\alpha)\vec{x}=\hat{n}(\hat{n}\cdot\vec{x})+\cos\left(\alpha\right)(\hat{n}\times\vec{x})\times\hat{n}+\sin\left(\alpha\right)(\hat{n}\times\vec{x}) to index notation (forget about covariant and contravariant indices). My attempt...
  26. G

    Vector Calculus: Mesh Size and Size of Largest Rectangle

    Homework Statement I need to find a sequence of partitions , let's call it S of R=[0,1]x[0,1] such that as the number of partitions k→∞ , then limit of the area of the largest subinterval of the rectangle in the partition, denoted a(S) tends to 0, but the mesh size m(S) is a non-zero value...
  27. N

    Maximizing Sub-Rectangle Area in a Sequence of Partitions for a Unit Square

    Homework Statement Let R be the unit square such that R= [0,1] x [0,1] Find a sequence of partitions of R such that the limit as k ->inf of the area of the largest sub-rectangle of the partition (where k is number of partitions) goes to zero but the mesh size does not go to zero. Depicting the...
  28. N

    How to Find a Sequence of Partitions in Vector Calculus?

    Hi, I need help getting a start on this exercise. Let R be the unit square. Find sequence of partitions such that the mesh size goes to zero as sequence goes to infinity -> for this, is this just a series of sub rectangles whose respective areas shrink at the same rate with respect to one...
  29. D

    Understanding the Chain Rule in Vector Calculus for Gradient of Scalar Functions

    Hi. I was looking for a chain rule in vector calculus for taking the gradient of a function such as f(A), where A is a vector and f is a scalar function. I found the following expression on wikipedia, but I don't understand it. It's taking the gradient of f, and applying that to A, and then...
  30. 1

    Vector Calculus - Use of Identities

    Homework Statement By using a suitable vector identity for ∇ × (φA), where φ(r) is a scalar field and A(r) is a vector field, show that ∇ × (φ∇φ) = 0, where φ(r) is any scalar field. Homework Equations ∇×(φA) = (∇φ)×A+φ(∇×A)? The Attempt at a Solution I honestly have no idea how to even...
  31. M

    Vector Calculus, Multivariable Calculus, Linear Algebra

    I am in Calculus BC in high school right now and I am really enjoying it and have finished all of the material for the year and I have heard different things about what class comes after Calculus BC (I believe BC is equivalent to Calc 1 & 2 in college). I have heard the next class, aka Calc 3...
  32. R

    MHB Vector Calculus - Cylindrical Co-ordinates

    I have a question, to use cylindrical coordinates to find the volume of the ellipsoid ${R}^{2}+{3z}^{2}=1$. I know for cylindrical coordinates the Jacobian is $r$ so I have some integral: $$\iiint (r)dzdrd\theta$$ However I am struggling to work out the bounds of the integral for $z,r,\theta$...
  33. NuclearMeerkat

    Verification of Stoke's Theorem for a Cylinder

    Homework Statement Homework Equations Stoke's Theorem: The Attempt at a Solution ∇×A = (3x,-y,-2(z+y)) I have parametric equation for wall and bottom: Wall: x(θ,z) = acosθ ; y(θ,z) = asinθ ; z(θ,z) = z [0≤θ≤2π],[0≤z≤h] Bottom: x(θ,r) = rcosθ ; y(θ,r) = rsinθ ; z(θ,r) = 0 [0≤θ≤2π],[0≤r≤a]...
  34. C

    Understanding Vector Calculus & Trigonometry for Physics

    Hi. Every time i read physics I feel so discouraged by my lack of understanding in terms of vector calculus and trigonometry. I do understand basic trigonometry and basic vectors but when professors or textbooks move onto combining derivatives with vectors i lack understanding of which rules...
  35. M

    Find a Rigorous Calculus of Multiple Variables Book for Advanced Learners

    Hi! I am looking for a very rigorous book on some of the topics covered in Calculus of Multiple Variables. My University uses the last part of Adams "Calculus: a complete course" and I found the presentation therein more fit for people needing to know enough to perform the calculations than for...
  36. S

    Can L be conserved if its magnitude is conserved?

    Problem: Consider a system for which Newton's second law is $$ \frac {d \vec v}{dt} = - [ \frac {h(r)h'(r)}{r} + \frac {k}{r^3} ] \vec r- \frac {h'(r)}{r} \vec L $$ where k is a constant, h(r) is some function of r, h'(r) is its derivative and L = r x v is the angular momentum. Show that $$...
  37. T

    Proof of equivalence between nabla form and integral form of Divergence

    Does anybody knows how you can reach one form of the divergence formula from the other? Or in general, why is the equivalence true?
  38. PhysicsKid0123

    Vector calculus, surfaces, and planes.

    I have attached an image... Okay, so I have been stuck on this problem for like 2 hours now and I have no idea how to find r(x). I know the trace is the intersection of the plane and the surface. My first attempt was to substitute the plane y+2x=0 equation for the surface equation by solving...
  39. B

    Proving Vector Calculus Identities: Tips and Tricks

    Homework Statement div(øu) = ødivu + ugradø Homework Equations divergence of scalar field = f,ii divergence of vector field = ui,i The Attempt at a Solution I've heard this is a simple proof, but this is my first one of 8 or so proofs I need to complete for homework, and I'm...
  40. J

    Vector Calculus Identites Question

    Homework Statement let r (vector) =xi+yj+zk and r=sqrt(x^2+y^2+z^2), let f(r) be a C2 scalar function. 1. Prove that ∇f = dr/df * vector r 2. Using part 1, calculate ∇ cosh(r^5), check answer by direct calculation 3. Using Vector Identities, calculate ∇ X (cosh(r^5)*∇f...
  41. M

    Best practice books on Vector Calculus

    I'm doing an undergraduate course in engineering 1st sem. I have physics which contains cumbersome amount of vector calculus operations. Although I have cleared my concepts I require a lot of practice to increase agility in vector calculus. I demand thus a good practice book or set of problems...
  42. T

    Vector Calculus Question in Lagrangian Mechanics

    Hi guys. I hope this isn't a bad place to post my question, which is: I'm reading some lecture notes on Lagrangian mechanics, and we've just derived the Euler-Lagrange equations of motion for a particle in an electromagnetic field. It reads: m \ddot{\vec{r}} = -\frac{e}{c} \frac{\partial...
  43. Z

    Parametrization of a Parallelogram: Mapping Rectangles onto Planar Regions

    Homework Statement (a) Show that the four points r1 = (1, 0, 1), r2 = (4, 3, 5), r3 = (6, 4, 6) and r4 = (3, 1, 2) are coplanar and the vertices of a parallelogram. Let S be the closed planar region given by the interior and boundary of this parallelogram. An arbitrary point of S can be...
  44. I

    Vector Calculus - Laplacian on Scalar Field

    A scalar field \psi is dependent only on the distance r = \sqrt{x^{2} + y^{2} + z^{2}} from the origin. Show: \partial_{x}^{2}\psi = \left(\frac{1}{r} - \frac{x^{2}}{r^{3}}\right)\frac{d\psi}{dr} + \frac{x^{2}}{r^{2}}\frac{d^{2}\psi}{dr^{2}} I've used the chain and product rules so...
  45. U

    Vector calculus identities proof using suffix notation

    I must become good at this ASAP. Homework Statement prove \vec{\nabla}\cdot (\vec{a}\times\vec{b} ) = \vec{b} \cdot(\vec\nabla\times\vec{a}) - \vec{a}\cdot(\vec\nabla\times\vec{b}) Homework Equations \vec a \times \vec b = \epsilon_{ijk}\vec a_j \vec b_k \vec\nabla\cdot =...
  46. L

    Is Vector Calculus Needed for Statistics?

    I've taken a multi-variable calculus course already that covers infinite sequences and series, Taylor's theorem, quadratics surfaces, double and triple integration etc. I'm looking to get a Master's Degree in statistics two years from now, is there any point of me taking a class that involves...
  47. N

    Can Vector Calculus Verify This Identity?

    Homework Statement Use your knowledge of vector algebra to verify the following identity: \vec{\Omega} \cdot \nabla n = \nabla \cdot \vec{\Omega} n Homework Equations Divergence product rule \nabla \cdot (\vec{F} \phi) = \nabla (\phi) \cdot \vec{F} + \phi (\nabla \cdot \vec{F})...
  48. P

    Understanding Vector Calculus Proof: Divergence Theorem and Scalar Field

    The following is used as part of a proof I'm trying to understand: ∫Vf(∇.A)dV=∫SfA.dS-∫VA.(∇f)dV where f is a scalar field, and the surface integral is taken over a closed surface (which presumably encloses the volume). I'm not sure how to go about proving this. I can see the divergence...
  49. M

    Vector calculus: angular momentum operator in spherical coordinates

    Note: physics conventions, \theta is measured from z-axis We have a vector operator \vec{L} = -i \vec{r} \times \vec{\nabla} = -i\left(\hat{\phi} \frac{\partial}{\partial \theta} - \hat{\theta} \frac{1}{\sin\theta} \frac{\partial}{\partial \phi} \right) And apparently \vec{L}\cdot\vec{L}=...
  50. J

    Rotational in terms of vector calculus

    Hellow! I was noting that several definitions are, in actually, expressions of vector calculus, for example: Jacobian: \frac{d\vec{f}}{d\vec{r}}=\begin{bmatrix} \frac{df_1}{dx} & \frac{df_1}{dy} \\ \frac{df_2}{dx} & \frac{df_2}{dy} \\ \end{bmatrix} Hessian: \frac{d^2f}{d\vec{r}^2} =...
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