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. 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...
  2. 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...
  3. 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...
  4. Y

    Vector Calculus: Find Line Perp. to L Through P in 6x-4y+2z=1 Plane

    Homework Statement 1. Let L (line) be the line given by x = 6 - t, y = 4 + t , z= 4 + t. L intersects the plane 6x - 4y + 2z = 1 at the point P = (6, 4, - 8). Find parametric equations for the line through P which lies in the plane and is perpendicular to L I don't even know where...
  5. S

    Center of mass and a bit vector calculus

    Hi guys, recently I am reading Kleppner's Mechanics about center of mass, well, I am always a fairly fast learner, but I really got stick here. you see, in the page 119 example 3.3 (in short, a rod with nonuniform density ) let Q(x) be the function density of location vector it said...
  6. Q

    Vector Calculus Boat Traversing a river problem

    Homework Statement Consider a long straight river flowing north with parallel banks 40m apart. Let us use the function u(x) = 3 sin(\pix/40) to model the rate of water flow x units from the west bank.A boat proceeds at a constant speed of 5m/s from a point A on the west bank while maintaining...
  7. E

    Understanding Velocity and Vector Calculus for Moving Objects

    Homework Statement An object moves according to this : r(t)= (1+t^2)i + (1-t)j + (t+t^3)k How fast does it moves far from O(0,0) for t=1? Is this velocity the same with the object's velocity? The Attempt at a Solution I can't understand the second part
  8. N

    Vector Calculus World Problem About Mechanics

    Homework Statement A particle is constrained to move around the unit circle in the xy plane according to the formula (x,y,z)=(cos(t2),sin(t2),0), t\geq0. At what point on the circle should the particle be released to hit a target at (2,0,0)? Homework Equations None The...
  9. M

    Vector Calculus: Calculate Work via Line Integral from (-1,2) to (1,3)

    Homework Statement A force is applied to a particle, defined by: F(x,y)= (y^2, 2xy) << This is a verticle bracket with the y^2 ontop of the 2xy The path of the particle is straight. The particle moves from (-1,2) to (1,3) i) Calculate the work that the force F does as the...
  10. S

    Vector Calculus - Divergence theorem

    Homework Statement 1. Consider a cube with vertices at A=(0,0,0) B=(2,0,0) C=(2,2,0) D=(0,2,0) E=(0,0,2) F=(2,0,2) G=(2,2,2) H=(0,2,2) A)Calculate the flux of the vector fieldF=xi through each face of the cube by taking the normal vectors pointing outwards. B)Verify Gauss's divergence theorem...
  11. N

    Vector Calculus - Closest point to the origin

    Homework Statement A curve C in space is defined implicitly on the cylinder x^2+y^2=1 by the additional equation: x^2-xy+y^2-z^2=1. Find the point or points on C closest to the origin.Homework Equations d = ((x-x0)+(y-y0)+(z-z0))^(1/2) - This is the distance formula. Please note that I did...
  12. L

    Teaching Myself Vector Calculus, need some advice.

    Hello, I am a physics major currently finished with my second year and I am trying to teach myself vector calculus since the Calculus III class at my university did not include it and I am taking upper level electromagnetic physics courses at another univeristy this coming up fall semester. So...
  13. T

    Vector Calculus II: Divergence

    Homework Statement A smooth vector field F has divF(1,2,3) = 5. Estimate the flux of F out of a small sphere of radius 0.01 centered at the point (1,2,3). Homework Equations Cartesian Coordinate Definition of Divergence: If F= F1i + F2j +F3k, then divF=dF1/dx + dF2/dy + dF3/dz The...
  14. T

    Vector Calculus II: Flux Integrals

    Homework Statement Calculate the flux of the vector field through the surface: F = i + 2j through a square of side length 2, lying in the plane x + y + z = 1, oriented away from the origin. Homework Equations The Attempt at a Solution So, I need to take the integral over the...
  15. T

    Vector Calculus II: Flux Integrals

    Homework Statement F = 2i + 3j through a disk of radius 5 in the plane y = 2 oriented in the direction of increasing y. Calculate the flux of the vector field through the surface. Homework Equations The Attempt at a Solution I know that I need to calculate the area vector...
  16. T

    Help with Vector Calculus practice test?

    Homework Statement Can anyone help me solve couple(if not all of these problems on this practice test? I have this huge test thursday and I can't seem to get any of these concepts thru my head. You might not be able to solve the first set of problems(1-6) because of a different teachers...
  17. M

    Vector Calculus Identities: Proving v · ∇v = ∇(0.5v2 + c × v)

    Homework Statement Show that v\nablav = \nablaxvxv v · ∇v = ∇(0.5v2 + c × v c=∇ × v My attempt ∇(A · B)= B · ∇A + A · ∇B + B×(∇×A) + A×(∇×B) Replace A and B with V ∇(v · v)= v · ∇v + v · ∇v + v×(∇×v) + v×(∇×v) v · ∇v = ∇(0.5v2 - v×(∇×v) Is v×(∇×v) = =∇ × v × v? And...
  18. S

    Finding the Second Force and Angle in Cartesian Vector Calculus

    Homework Statement Two forces act on an object at an angle of 50°. One force is 150 N. The resultant force is 200 N. Find the second force and the angle that it makes with the resultant, using only cartesian vectors. Homework Equations The Attempt at a Solution Over here, I am very confused...
  19. S

    How Do You Calculate Distance and Displacement in Vector Calculus?

    Homework Statement The displacements of two ships, A and B, two hours after leaving from the same port can be represented with position vectors \vec{OA} [20, 50, 0] and (\vec{OB}) [60, 10, 0]. Assume that the port is located at the origin and that all units are in kilometres. a. How...
  20. U

    Exploring Curl: A Closer Look at the Circulation of Vector Fields

    I realize by now I must be making everyone crazy on this forum with my questions on vector calculus...but I really have no choice! Please bear with me for another few days...I promise I'll get this done as fast and as painlessly as possible. Here goes... 1.In divergence theorem,green's...
  21. D

    Line integral, vector calculus

    Evaluate the line integral Force field is the integral in the form of integrand ( (2x dx+ 2y dy + 2 zdz)/r^2). the domain of integral is C, C = C1 + C2. C1 is the line segment from (1; 2; 5) to (2; 3; 3). C2,arc of the circle with radius 2 and centre (2; 3; 1) in the plane x = 2. The...
  22. A

    How hard of an assignment does this look like for Vector Calculus?

    My course is supposed to go from basic vector algebra to extrema, curl and divergence, (stopping before multiple integration) I don't need any help on this assignment I am just wondering how hard it looks compared to everything else. http://tinypic.com/r/2ly29u8/6
  23. R

    What are the best books for understanding vector calculus and analysis?

    Can u suggest a me a really good book on vector calculus and analysis , the book should be more of conceptual nature , because i want to know the indepth concept and meaning of various things like curl, grad, div, various theorems. I mean book shouldn't be problem oriented. I don't need too much...
  24. E

    Vector calculus: divergence of a cross product

    Homework Statement I need to prove the identity div (a x b) = b dot (curl a) - a dot (curl b) The Attempt at a Solution I've done the proof about 10 times now, and everytime I get the left hand of the identity equal to this: (all the d's are partial derivatives) d(a3b1)/dx -...
  25. X

    Oversimplified Vector Calculus Problem

    I'm not asking for a solution to this problem. I'm just wondering about its validity. Homework Statement The wind velocity v1 is 40mph from east to west while an airplane travels with air speed v2 of 100mph due north. The speed of the airplane relative to the ground is the vector sum v1...
  26. H

    How Does the Index in This Vector Calculus Identity Work?

    Homework Statement I'm a bit confused as to the following vector calculus identity: [∇ (∇.A)]_i = (δ/δx_i )( δA_j/δx_j) Shouldn’t it be = (δ/δx_i )( δA_i/δx_i) why is it ‘j’ if we are taking it over ‘i’ ? Thanks.
  27. H

    What are some useful vector calculus identities for vector fields A and B?

    Homework Statement For arbitrary vector fields A and B show that: ∇.(A ∧ B) = B.(∇∧A) - A.(∇∧B) The Attempt at a Solution I considered only the 'i'-axis, by saying that it is perpendicular with A and B and then I expanded both the left and right side out. The working is...
  28. M

    Solving Vector Calculus: (a+2b)∇(∇⋅u) - b∇x∇xu - (3a+2b)c∇T(r)=0

    I'm unsure how to do this problem: (a + 2b)\nabla(\nabla \cdot \vec u) - b \nabla \times \nabla \times \vec u - (3a + 2b)c\nabla T(r)= \vec 0 \hat u = U_r \hat r + u_\theta \hat \theta +u_z \hat z a,b,c constants how would I solve this for u?
  29. S

    Surface area of a hemisphere w/ vector calculus

    Homework Statement I need help proving how you could use evaluation of the surface integral \oint\oint f(x,y,z)dS to show that the surface area of the upper hemisphere of radius a is 2\pi a2. So any ideas? Homework Equations The teacher mentioned that the divergence theorem would be...
  30. R

    Recommend a Vector Calculus book

    Recommend a "Vector Calculus" book - by "Vector Calculus" i do not mean Calc III (multivariate). What i mean is the Calculus course that follows after Calc III and consists of topics such as Stokes's and divergence theorems, Cartesian tensors. - Wel'll be using Vector Calculus by Jerrold E...
  31. A

    Vector calculus fundamental theorem corollaries

    Homework Statement Prove \int_{V}\nabla\ T d\tau\ = \oint_{S}Td\vec{a} Homework Equations Divergence theorem: \int_{V}(\nabla\bullet\vec{A})d\tau\ = \oint_{S}\vec{A}\bullet\ d\vec{a} The Attempt at a Solution By using the divergence theorem with the product rule for...
  32. O

    How to Prove the Identity ∇×(u×v)=v∙∇u-u∙∇v+u∇∙v-v ∇∙u?

    Can any show me how you will go about proofing this identity ∇×(u×v)=v∙∇u-u∙∇v+u∇∙v-v ∇∙u where v and u are vectors Many thanks.
  33. R

    One last vector calculus problem

    Homework Statement Consider a hill that can be modeled by the function z=\frac{14-x^2-y^2}{3} where the +ve x-axis points south and the +ve y-axis points east. Person A is standing at P=(1,2,3) on the hill. i) if A walks due west from P, does he ascend or descend the hill and at what...
  34. R

    Sketch Cone z2=x2+y2 in 3D Space + Vector Calc

    Homework Statement Sketch the cone z2=x2+y2 in 3D space. Let (x0,y0,z0)≠(0,0,0) be a point on the given cone. By expressing the fiven equation of the cone in the form f(x,y,z)=a, find a normal vector tot he cone at point (x0,y0,z0) Find the equation of the tangent plane to the cone at...
  35. R

    Vector calculus identity proof.

    Homework Statement Let f(x,y,z) be a function of three variables and G(x,y,z) be a vector field defined in 3D space. Prove the identity: div(fG)= f*div(G)+G*grad(f) Homework Equations For F=Pi +Qj+Rk div(F)=dF/dx + dQ/dy + dR/dz grad(F)=dF/dx i + dQ/dy j + dR/dz k The...
  36. J

    How Do I Use Unit Directional Vectors in Vector Calculus?

    Homework Statement Attached Homework Equations The Attempt at a Solution I'm not quite sure how to start these. Once I figure out one of them, I should be able to do the rest. The confusing piece to me is with the unit directional vector (a_rho etc. My teacher has me horribly...
  37. Vectronix

    Vector Calculus: Question about the origin of the term 'divergence'

    Why is the divergence operation called the 'divergence?' What is the significance of this operation on a vector-valued function? And what about "the curl?" The curl seems self-explanatory (at least it does in electrodynamics), but I need someone to expound on 'the curl' as well.
  38. M

    I would like a hint on how to begin a vector calculus problem

    Homework Statement I am just confused about how the question is structured, and I am unsure on how to get the relevant information to answer the question: Find the angle between the surface normal directions of r^{2} = 9 and x + y + z^{2} = 1 at the joined point (2,-2,1) Homework...
  39. T

    Electromagnetism( Vector Calculus)

    Homework Statement Any sites you recommend that can really supplement my understanding of Vector Calculus applied to Electromagnetis? Thanks
  40. D

    Vector calculus identities proof

    Hello. How can I prove something like \nabla\cdot(\mathbf fv)=(\nabla v)\cdot\mathbf f+v(\nabla\cdot \mathbf f) using only the definition of divergence \text{div}\mathbf V=\lim_{\Delta v\rightarrow0}\frac{\oint_S\mathbf V\cdot d\mathbf s}{\Delta v}, i.e. without referring to...
  41. B

    Looking for an electrodynamics text that includes vector calculus instruction

    Hello everyone, I wonder if a textbook like the one I described in the subject line exists - most of the classical electrodynamics textbooks I've looked at assume you already have a solid background in vector calculus. I'm trying to do a self-study in electrodynamics, and while I have taken...
  42. L

    Vector calculus in higher dimensions

    in 2d, "curl"(grad(f)) = 0 "curl" is the operation that green's tehorem talks about. In 3D, curl(grad(f)) = 0 and div(curl(F)) = 0. We may consider vector calculus in 4 spatial dimensions, for vector fields F:R^4 -> R^4. what is "curl" like in 4D, since curl is actually only difined in 3D...
  43. P

    Vector Calculus: Meaning of Volume Int. of Grad-T

    Hello, I'm working through some problems in the Griffith text on electrodynamics. In one of them, the reader is asked to prove the following identity (which is given in the text), which is a generalization (of sorts) on the divergence theorem: \large{ \int_V \left(\nabla T\right) dV =...
  44. M

    Surface integral from vector calculus

    Homework Statement F = (x^2) i + (y^2) j + (z) k, S is the cone z = (x^2+y^2) ^ (1/2), with x^2 +y^2 <= 1, x >= 0, y >= 0, oriented upward. Homework Equations All of the above The Attempt at a Solution My attempted solution is 0. But other students claim that the answers is...
  45. M

    Vector Calculus: Surfave Integrals

    Vector Calculus: Surface Integrals Homework Statement Find the surface integral of u[/B dot n over S where S is part of the surface z = x + y^2 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 The Attempt at a...
  46. L

    Vector Calculus: Is F=0 When ∇·F=0 & ∇×F=0?

    is it true that if \nabla \cdot \vec{F}=0 , \nabla \times \vec{F}=0 then \vec{F}=0?
  47. H

    Is this vector calculus notation correct?

    My vector calculus is a bit rusty. Can anyone tell me if the following uses proper symbolism? F &= \left[\begin{matrix}f_1(x_1,x_2) \\ f_2(x_1,x_2) \\ f_3(x_1,x_2) \end{matrix}\right] \qquad x = \left[\begin{matrix} x_1 \\ x_2 \end{matrix}\right] \qquad \frac{DF}{dx}&=...
  48. M

    Visualizing Vector Calculus: Sketching Field Lines with Scalar Functions

    Homework Statement Let r be a position vector from the origin (r=xi+yj+zk), whose magnitude is r, and let f(r) be a scalar function of r. Sketch the field lines of f(r)r 2. Homework Equations 1 \nablax(\nabla\Psi)=0 2 \nabla.(\nablaxv)=0 3...
  49. A

    Evaluating Vector Calculus Line Integral

    use green lemma to evaluate line integral int of (x^2-y^2)dx-4xydy over rgn bounded by y^2=4x,y=0,x=1 in ist quadrant. attempt double integral (dv/dx -du/dy)dx dy=int. from limit 0to 2(int.from 0to 2x^1/2(6ydxdy)) =int.lim 0to 2(12x^3/2)dy =24/5(2^5/2) ans in book is -2.
  50. F

    Can You Subtract Vectors in Spherical Coordinates to Find Distance?

    Hi all, Would someone please re-enlighten me. Say I have a vector in spherical coordinates: \vec r_1 = \phi \hat{\phi} + \theta \hat{\theta} + R \hat{R} Where r, \theta, R are scalars and the corresponding hat notation is the unit vectors. Say, I form a new vector r_2 in...
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