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. skate_nerd

    MHB Proving vector calculus identities w/ tensor notation

    I have an vector calculus identity to prove and I need to use vector notation to do it. The identity is $$\vec{\nabla}(fg)=f\vec{\nabla}{g}+g\vec{\nabla}{f}$$ I tried starting with the left side by writing $\vec{\nabla}(fg)=\nabla_j(fg)$. Now I look and that and it really looks like there is...
  2. M

    Vector Calculus: Prove ∇ × (a×r/r^n) = (2-n)a/r^n + n(a.r)r/r^(n+2)

    Homework Statement Prove that ##∇ × (\frac{\vec{a} × \vec{r}}{r^n})= \frac{2-n}{r^n}\vec{a}+\frac{n(\vec{a}.\vec{r}) \vec{r}}{r^{n+2}}## Nothing is mentioned about ##\vec{a}## so I'm assuming it is a constant vector. Also ##\vec{r} = <x, y, z> ## and ##|r|=r## Homework Equations...
  3. G

    Vector calculus: how to order terms?

    Hi, I need to let an operator act on a scalar function. The operator is however in a very cryptic form, so I would want to work it out a little bit. I get stuck in the process. The operator is: \vec{u}\cdot\left[\vec{L}\times\left(\vec{u}_r\times\vec{L}\right)\right]f Where \vec{L} is...
  4. P

    Vector calculus question on showing the area of a surface is infinite

    Homework Statement Let S be the surface z = 1/(x^{2} + y^{2})^{1/2}, 1 ≤ z < ∞. Show that the area of S is infinite. Homework Equations the surface S is given by z=f(x,y) with f(x,y)=1/(x^{2}+y^{2})^{1/2} and for x,y in the disk D which is the circle seen when the surface is viewed from the...
  5. K

    I'm looking for a vector calculus book

    I'm looking for a vector calculus book that's good for self study. I need to get some extra practice would like something with a good number of exercises covering all the regular vector calculus topics. A book as applied as possible would be nice. Any suggestions? I've used Vector Calculus by...
  6. T

    Vector Calculus Proof: Curl V = 0 -> V = grad phi

    If ∇ x v = 0 in all of three dimensional space, show that there exists a scalar function ##\phi (x,y,z)## such that v = ∇##\phi##. (from Walter Strauss' Partial Differential Equations, 2nd edition; problem 11; pg 20.) I'm not really sure where to begin with this problem. I asked a few of my...
  7. C

    Can this simple trig problem be solved using vector calculus?

    I recently have been teaching myself vector calculus online, i am by no means a master but i get the general concepts. I know you can use it to solve the motion of a particle in a fluid and was curious as to whether it can be used to solve simple physics problems, involving current and wind...
  8. D

    MHB Fernet-Serrat equations and vector calculus

    I have shown the first two equality and I am working on the showing the 1st equals the 3rd. \begin{alignat*}{4} \frac{1}{\rho}\hat{\mathbf{{n}}} &= \frac{d\hat{\mathbf{{u}}}}{ds} &{}= \frac{\dot{\hat{\mathbf{{u}}}}}{\dot{s}} &{}= \left((\dot{\mathbf{r}}...
  9. M

    Any good books in Vector Calculus?

    I have been restudying vector calculus, especially on topics pertaining to line integrals, surface integrals (and the accompanying vector forms). One problem I have encountered from the book I have been using is that it seems there are some theorems and results that are only restricted to...
  10. F

    I have a cumbersome problem with Vector calculus

    I am unfamiliar with Vector calculus, a tool for learning Physics I select a homework I did not solve yet, then hope a help from you guys, in attachment pdf file My attempt: I tried to use BAC-CAB rule, but the key hardness of mine is I still do not know the concepts clearly (as you know a...
  11. B

    Vector Calculus Theorems - Duality Question

    I'm trying to go over some vector analysis using forms & kind of noticed what look like random vector identities are more appropriately thought of, to me at least, as differential analogues of the classical integral theorems in the way Maxwell's equations can be cast in differential & integral...
  12. J

    Vector calculus, Torricelli's Trumpet/Gabriel's Horn

    Homework Statement I've recently been completing an assessment on Torricelli's Trumpet and was told to look into the geometry aspect. I've been following this website: http://www.palmbeachstate.edu/honors/documents/jeansergejoseph.pdf I understand all the steps but am not privvy to how they...
  13. M

    Maximizing an evolutionary biology equation (vector calculus)

    Homework Statement For a Gaussian landscape, the log-fitness change caused by a mutation of size r in chemotype element i is Q_i(r) = -\vec{k} \cdot S \cdot \hat{r_i}r - \dfrac{1}{2} \hat{r_i} \cdot S \cdot \hat{r_i}r^2 . To find the largest possible gain in log-fitness achievable by...
  14. Y

    Help with vector calculus in reflection and transmission of plane wave

    This is not a homework, this is concerning reflection and transmission of electromagnetic wave ( plane wave) at a flat planar boundary between two media. But the work in question is pure vector calculus. I ultimately want to proof if ##\vec E_I=\hat y E_I## then ## \vec E_R## and ##\vec E_T##...
  15. D

    Vector calculus for ellipse in polar coordinates

    Hello =] I'm having trouble with this question, can somebody please help me with it! I'll thanks/like your comment if help me =) ![Question][1] I know that for a ellipse the parametric is x=a sin t , b= b cos t t:0 to 2pi (?) for part a) I drew up the graph but not sure if it's...
  16. O

    Vector calculus -line integral

    Vector calculus ---line integral Homework Statement If vector F(r)= (x^2)y i + 2yz j + 9(z^2)x k , find ∫ vector F dot vector dr between (0,0,0) and (1,2,3) Homework Equations The Attempt at a Solution If I want to find work done, then I just use F dot dr in this case , in F act...
  17. J

    Is Vector Calculus useful for pure math?

    Hi everyone, I have the option to take a vector calculus class at my uni but I have received conflicting opinions from various professors about this class's use in pure math (my major emphasis). I was wondering what others thought about the issue. I appreciate any advice.
  18. N

    Confused about Vector Calculus Curvature Formulas? Let's Clear Things Up!

    Hey. so you have two formulas for curvature: The ordinary: |dT/ds| = |a|/|v|2 And the advanced: |v x a|/|v|3 = |a|*sin(α)/|v|2 = |aN|/|v|2 But the problem is, those two formulas aren't the same? The top one has acceleration divided by speed squared, while the bottom one has normal component...
  19. Y

    Can the Chain Rule be Applied to Show the Identity in Vector Calculus Homework?

    Homework Statement \widetilde{F}(r)=F1(r)i+F2(r)j+F3(r)k \hat{r}=r/r r(x,y,z)=xi+yj+zk, r=abs(r)=sqrt(x2+y2+z2) (Hint: The chain rule will be helpful for this question.) Show that: \nabla\cdotF = \hat{r}\cdotdF/dr. Homework Equations The Attempt at a Solution My attempt...
  20. M

    Vector calculus and parametrisation

    Ok, I just found out I have a physics assignment due tomorrow and I have no idea how to do it so I came here for help as none of the maths assistants at Uni could help me. I'm having trouble with: 1. Consider the parametric curve given by the equation x(t)= t<i> + t^(1/3)<j> - <> denotes a...
  21. D

    Vector Calculus - gradient geometry

    Hello. I can't seem to wrap my head around the geometry of the gradient vector in ℝ3 So for F=f(x(t),y(t)), \frac{dF}{dt}=\frac{dF}{dx}\frac{dx}{dt}+\frac{dF}{dy}\frac{dy}{dt} This just boils down to \frac{dF}{dt}=∇F \cdot v Along a level set, the dot product of the gradient vector and...
  22. M

    Vector Calculus by Marsden and Tromba vs. Vector Analysis by Brand

    Hi. I have heard that the marsden and Tromba book is not very rigorous. Is this true? I seek a vector book that is something proof intensive. I took vector analysis b4 with a book called vector calculus by Susan colley, I am looking for something more rigorous than this. Anyone read vector...
  23. J

    Projection of surface area elements in vector calculus

    Homework Statement (i) Find the normal, n, at a general point on the surface S1 given by; x2+y2+z = 1 and z > 0. (ii) Use n to relate the size dS of the area element at a point on the surface S1 to its projection dxdy in the xy-plane. The Attempt at a Solution To...
  24. micromass

    Calculus Vector Calculus, Linear Algebra, and Differential Forms by Hubbard

    Author: John Hubbard, Barbara Hubbard Title: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach Amazon Link: https://www.amazon.com/dp/0971576653/?tag=pfamazon01-20
  25. P

    Interpreting and Solving a Vector Calculus Question

    Hi - I'm totally stuck with this question: how to interpret it and tackle it. Any advice woiuld be greatly received! We've not covered anything like this in classes... Let A = \left( x_{A}, y_{A}, z_{A} \right) B = \left( x_{B}, y_{B}, z_{B} \right) be two given distinct points in the...
  26. H

    Does Vector Calculus Allow Operators to Function Like Vectors?

    Reading the Feynman Lectures, \nabla \times (\nabla T)=(\nabla \times \nabla) T, is achieved by analogy to the analogous case for \mathbf{A} \times (\mathbf{A} T)=(\mathbf{A} \times \mathbf{A}) T,where T is a scalar field in all cases. While this is obvious if \nabla were to be replaced...
  27. I

    A little help with vector calculus

    How can I describe electric fields and equipotential surfaces using Vector calculus?? HELP...
  28. C

    Vector Calculus with Maxwell's Equations

    Homework Statement Consider the following representation of Maxwell's eqns: $$\nabla \cdot \underline{E} =0,\,\,\, \nabla \cdot \underline{B} = 0,\,\,\, \nabla \times \underline{E} = -\frac{\partial \underline{B}}{\partial t}, \,\,\,\frac{1}{\mu_o}\nabla \times \underline{B} = \epsilon_o...
  29. O

    Description of surface, vector calculus

    Homework Statement Consider the surface parameterized by (v cos(u), v sin(u), 45 v cos(u)), where u and v both vary from 0 to 2∏.Homework Equations (v cos(u), v sin(u), 45 v cos(u)) I think this is supposed to be a vector function? As in r(u,v) = <v cos(u), v sin(u), 45 v cos(u)>.The Attempt at...
  30. P

    A vector calculus proof question

    Homework Statement The image contains the problem statement and all relevant equations. I have no idea what to do, this is all very new to me...plus the hint doesn't make sense to me. http://s13.postimage.org/b0kq27bkn/photo_6.jpg photo uploader
  31. B

    Vector Calculus Problem - Griffith Textbook

    I have a problem in the Griffith textbook (Introduction to Electrodynamics), Question 1.21, where it asks what is the meaning of the vector (A.∇)B, my simplistic approach would be to calculate the divergence of A which should be a scalar and multiply it out by Bx,By,Bz) to compute the x...
  32. C

    Vector Calculus Question about Surface Integrals

    Why is it that when the force field is z^2 and you take the surface integral over a sphere of radius a using spherical coordinates, that yields the flux to be (4pi a^3 )/ 3 BUT in a calculus book, the force field is z instead of z^2 evaluated using polar coordinates and it yields the same...
  33. G

    A Vector Calculus Identity for Characteristic Projections in PDEs

    In the notes it says that \text{v}\cdot \nabla \text{u} = |\text{v}|\frac{du}{dl} \text{v} = (a(x,y), b(x,y)) l is the arclength in the v-direction. Why is this? The LHS is the projection of v onto the gradient of u, the other thing is the magnitude of v, multiplied by the du/dl.
  34. Kushwoho44

    Vector Calculus: Area and Mass of a Region

    Hi y'all. Here is exactly what is stated on the theory page of my book: Example: Area of a Region The area of a region R in the xy-plane corresponds to the case where f(x,y)=1. Area of R= ∫∫dR Example: Mass of a Region The mass of a region R in the xy-plane with mass density per unit area...
  35. M

    Prove the functions are unique in a volume, vector calculus problem

    Homework Statement In a volume V, enclosed by a surface S, the vector fields X and Y satisfy the coupled equations ∇×∇×X=X+Y ∇×∇×Y=Y−X If the values of ∇×X and ∇×Y are given on S, show that X and Y are unique in V. Homework Equations ∇.(A×B)=B.(∇×A)−A.(∇×B)...
  36. S

    How do Physicists apply Vector Calculus to Physics?

    So, how do people, like physicists, or engineers, actually apply the concepts of vector calculus to their work. For example, if they want to calculate flux or something they need a vector field, how do they approximate that vector field? are their specific equations that can approximate vector...
  37. A

    Schools Should I take Vector Calculus before General University Physics?

    I'm currently in my first semester at a community college with intention of transferring to a 4 year. At some point while still at my current school I will have to take General University Physics. My curriculum is saying that I should take both Uni. Physics and Vector Calculus in my third...
  38. G

    Electrodynamics and vector calculus question

    Homework Statement 1) The magnetic field everywhere is tangential to the magnetic field lines, \vec{B}=B[\hat{e}t], where [\hat{e}][/t] is the tangential unit vector. We know \frac{d\hat{e}t}{ds}=(1/ρ)[\hat{e}][/n] , where ρ is the radius of curvature, s is the distance measured along a...
  39. Avatrin

    Complex Analysis and vector calculus

    Hi How much different is complex analysis from vector calculus? To me complex analysis looks like vector calculus combined with algebra of complex numbers..
  40. E

    Finding volume using the triple scalar product (vector calculus))

    Of the 3 vectors, does it matter what order I cross / dot them? <a \times b> \bullet c =? <a \times c> \bullet b
  41. H

    Deriving the Vector Calculus Equation for Magnetic Force

    I was reading a paper and came across this equation: Fmagnetic=μ0(M<dot>∇)H Is this the correct expansion below? (I'm not too experienced with vectors operating on the gradient operator) Fmagnetic=μ0[(Mx ∂H/∂x)i + (My ∂H/∂y)j + (Mz ∂H/∂z)k] _____________ My reasoning partially comes from...
  42. S

    Can anyone confirm this possible error in 'Vector Calculus' by Matthews?

    Homework Statement Find the surface integral of u dot n over S where S is the part of the surface z = x + y2 with z<0 and x>-1, u is the vector field u = (2y+x,-1,0) and n has a negative z component. Homework Equations In the text leading up to the end-of-chapter exercises (where this...
  43. B

    Vector calculus questions in electrodynamics

    im reading introduction to electrodynmics by griffiths, the math techniques used is sloppy to the point of frustration. hence i have several problems with the math while reading the text 1) it introduces the dirac delta function in dimension 1 δ(X) = 0 if x≠0 and δ(x)= ∞ if x= 0 and ∫δ(x)dx...
  44. A

    (Vector Calculus) Help regarding area element notation

    Homework Statement The area element of a sphere in spherical coordinates is given as following dA = r^2 \sin(\phi)\; d \theta \; d \phi using the notation in the following figure: However, while going through some E&M books I ran into the following notation Surface \; Area = r^2 \...
  45. A

    Question regarding vector calculus

    I have a question regarding vector calculus A particle P, Whose position vector is r=ti-(t+1)j+t^2k, moves along a curve. Draw curve "C" on -5<t<5 and write the: a) Parametric vector equation & symmetric equations of the tangent to the curve at (1,-2,1). b)Find all vector of t for which P...
  46. Z

    Very Basic Vector Calculus Question - dx,dy,dz and i,j,k

    I am learning about Stokes, Green's, and Gauss Divergence Theorems but from the angle of differential forms (the progression found in Pugh's "Real Mathematical Analysis"). This is supplemented by some more computational books, and I notice that these books frequently toss around i, j, and k...
  47. G

    Unique Solutions for Vector Calculus Problem with Boundary Conditions

    Homework Statement ## \gamma_1 ## and ##\gamma_2 ## are both real continuous solutions of ## \nabla^2 \gamma = \gamma ## in ## V## and ##\gamma_1=\gamma_2 ## on the boundary ##\partial V##. We are looking at the function ##g = \gamma_1 - \gamma_2 ##. I have proved ##\nabla \cdot \left( g...
  48. F

    Vector calculus question - surface of ellipsoid

    Homework Statement Let E be the ellipsoid \frac{x^2}{a^2}+\frac{y^2}{b^2}+z^2=1 where a>\sqrt{2} and b>\sqrt{2}. Let S be the part of the surface of E defined by 0\le x\le1, 0\le y\le1, z>0 and let \mathbf{F} be the vector field defined by \mathbf{F}=(-y,x,0). Given that the surface area...
  49. M

    Differential Forms and Vector Calculus

    So about a hundred years ago there was a live (sort of) differential forms thread hosted by someone named Lethe that was really helpful but short-lived. There have been some other diffl forms threads, too, such as the one centered on Bachman's book, but they all seem to peter out without any...
  50. L

    What does the ^ notation mean in vector calculus?

    Homework Statement I'm having problems with question 12b of the attached past exam paper, because I have no idea what the notation ^ means in vector calculus. If someone could explain that to me, I'd be really grateful. :-) Homework Equations The Attempt at a Solution .
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