What is Vector function: Definition and 62 Discussions

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the dimension of the function's domain is not defined by the dimension of the range.

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  1. O

    Arc length of vector function - the integral seems impossible

    The vector equation is ## v(x)=(e^x cos(2x), e^x sin(2x), e^x) ## I know the arc-length formula is ## S=\int_a^b \|v(x)\| \,dx ## I found the derivative from a previous question dealing with this same function, but the when I plug it into the arc-length function I get an integral that I've...
  2. 0

    Is the Intersection of Two Surfaces a Cylinder or Paraboloid in 3D?

    I'm given equations of surfaces and asked for the vector function that represents the intersection of the two surfaces. For ex: $$x^2 + y^2 = 4$$ and $$z = xy$$ In the solutions manual the answer is given like this: a sum of terms of cos t and sin t (is this polar form?). The way I did wasn't...
  3. F

    I Dimension of a set with vector function

    I have a vector ##\textbf{v} \in \mathbb{R}^{3N}## and a function ##\textbf{Ψ} : \mathbb{R}^{3N} \longrightarrow \mathbb{R}^p## such that ##\textbf{Ψ}(\textbf{v})=0##. Why the set ##T=\{ \textbf{x} \in \mathbb{R}^{3N} \ | \ \textbf{Ψ}(\textbf{x})=0 \}## has dimension ##n=3N-p##?
  4. M

    I Understanding Divergence of Vector Function F in 3D Space

    For the vector valud function F in the image, the three components of the output vector at a point are functions of (x,y,z)the three coordinates of the point.But while calculating divergence, why is the rate of change of x component of the output along x direction alone is accounted(similarly...
  5. DarkBabylon

    I Determining the flux of an arbitrary vector function

    Hello there. I've been working on trying to re-derive a certain physical formula using vector calculus, and came to a conclusion that in order to derive it, I'll need a way to determine the nature of a certain expression. Specifically: ∯f(v)·da - v={x1,x2,x3,...,xn} and f(v) returns a vector in...
  6. N

    I What is *not* a Vector Function?

    i never thought of it before! every function that I've encountered has been a 'vector function' .. so what is not a vector function?
  7. Sirsh

    Limiting Value of a Vector Function

    Homework Statement The position vector of a particle is given in the terms of t, by, s = (e-t+3*cos(2t)i+2tj+(e-t+3*sin(2t)k) Find the limiting value of speed when t approaches positive infinity. The answer says "s = ..." The Attempt at a Solution I have evaluated the limits of all the...
  8. Thales Costa

    I Plotting and visualizing a 3D plot of a vector function

    Hello, I'm trying to figure out how to plot a certain vector valued function but I'm having a hard time. The problem gives me the following vector valued function: r(u,v) = <u + v, 3 - v, 1 + 4u + 5v> I don't know how to plot this. So far I've tried making a table with some u and v values to...
  9. Thales Costa

    I Show that a particle moves over a circumference

    The problem asks me to show that a particle moves over a circumference with its center at the origin. The position vector of a moving particle is: I've tried using the x2+y2=r2 formula of the circumference, squaring both components of the vector function but I couldn't figure out what to do...
  10. H

    Calculus of vector function

    Homework Statement The position vector of a particle at time t is given by r(t)= 2sin(2t)i + cos(2t)j + 2tk where t >=0. Homework Equations v(t) = 4cos(2t)i - 2sin(2t)j + 2k speed = | v(t) | =√(16cos^2(2t)+4sin^2(2t)+4) = √(12cos^2(2t)+8) The Attempt at a Solution I found the velocity and...
  11. T

    Turning a vector into a vector function of time

    Homework Statement [/B] A Velocity vector: V = (12,4) write the vector as a vector function of Displacement. 2. The attempt at a solution I integrated the components of the Vector and got the function S(t) = (S(12t), S(4t)) I this correct at all?
  12. S

    Laplacian of a vector function

    Problem: The vector function A(r) is defined in spherical polar coordinates by A = (1/r) er Evaluate ∇2A in spherical polar coordinates Relevant equation: I'm assuming I have to use the equation 1671 on this website But I haven't got a clue as to how I would apply it since, for example, I...
  13. Yae Miteo

    What is the vector function for the intersection of a cone and a plane?

    Homework Statement "Find a vector function that represents the curve of intersection of the two surfaces." Homework Equations Cone: z = \sqrt{x^2 + y^2} Plane: z = 1+y The Attempt at a Solution I began by setting x=cos t, so that y = sin t and z = 1+sin t. At this point...
  14. I

    MHB Derivative of vector function

    $r(t)=\left\langle t-2, t^2+1 \right\rangle$, $t=-1$ sketch the plane curve with the given vector equation. $x=t-2$ and $y=t^2+1$ $x+2=t$ $(x+2)^2=t^2$ $(x+2)^2+1=t^2+1$ $(x+2)^2+1=y$ $x^2+4x+4+1=y$ $y=x^2+4x+5$ it's a parabola find $r'(t)$ $r'(t)=\left\langle 1, 2t \right\rangle$ sketch...
  15. I

    MHB Finding Vector and Parametric Equations for a Segment Joining Two Points

    Find vector and parametric equations for the segment that joins the points p(2,0,0) and q(6,2,-2). The examples in my book make no sense and i don't understand cheggs method. I know that r=(1-t)r_0 + t*r_1. I only need help finding the vector equation. Can someone give a step by step explanation?
  16. P

    Derivative of a Vector Function

    Homework Statement r(t) = ln ti + j, t > 0 find r′ (t) and r″(t)Homework Equations none The Attempt at a Solution r'(t)= 1/t i am I on the right track? The answer in the back is r'(t)= 1/t i -1/t^2 j Please help asap this is quite urgent! Thank you!
  17. D

    MHB Partial Derivatives of Matrix/Vector Function: An Easier Way?

    I was working on a pde, and I needed to compute a Jacobian for it. Suppose we have a function consisting of a series of matrices multiplied by a vector: f(X) = A * B * b --where X is a vector containing elements that are contained within A, b, and/or b, --A is a matrix, B is a matrix, and b is...
  18. D

    Using the product rule for the partial derivative of a vector function

    I was working on PDE for a project and needed to compute a Jacobian for it. Suppose we have a function consisting of a series of matrices multiplied by a vector: f(X) = A * B * b --where X is a vector containing elements that are contained within A, b, and/or b, --A is a matrix, B is a...
  19. J

    Calculate the arc length of the vector function

    Homework Statement Calculate the arc length of <2t,t^2,lnt> from 1=<t=<e Homework Equations Arc length=∫√{(x')^2 + (y')^2 + (z')^2} The Attempt at a Solution So I have gotten to this point: ∫√{4 + 4t^2 + \frac{1}{t^2}} Am I on the right track, and if so, how do I integrate that?
  20. H

    Linearly independence of vector function

    Given two vectors x(t) = (e^t te^t)^T y(t) = (1 t)^T a) Show that x and y are linearly dependent at each point in the interval [0, 1] b) Show that x and y are linearly independent on [0, 1] I compute det([x y]) = 0, so they are linearly dependent how about part b. Isn't a)...
  21. P

    Mathematica Differentiating vector function [Mathematica]

    Hi. So I have this vector function which I need to differentiate, it is however very tricky to do by hand, so I'm doing it in Mathematica. \hat{u}=\left\langle\bar{u}+\bar{r}\frac{(1+\gamma)}{r(r+\bar{u}\cdot \bar{r})}\right\rangle (The brackets denote normalisation) I want to do this...
  22. A

    Derivative of a vector function

    Hey, Can somebody help me on this one. I feel out of my depth and have to solve it somehow. I have a variable vector v=[v1 v2]T, a constant vector vc = [vc1 vc2]T, a scalar variable d and a vector function: s= d/(Vs/V-1) I need the first derivative ds/dv at a point of the mean of v...
  23. X

    Taylor expansion of a vector function

    Could someone please explain how does this taylor expansion work: 1/|r-r'| ≈ 1/r+(r.r')/r3 possibly you have to taylor expand twice to get this result, an attempt at which led me nowhere, surely it cannot be this complicated. any useful comment about this would be greatly appreciated...
  24. T

    How to find equation for a position vector function on a parabaloid?

    Homework Statement Velocity vector given by r'(t)=<cos(t), -sin(t), -2sin(t)> Surface that the position vector needs to lie on: z=x2+y2 Homework Equations Integral of r'(t) will give position function The Attempt at a Solution I know that the integral of r'(t) will give me...
  25. L

    Vector Function of Cone & Plane Intersection Curve

    Homework Statement Find a vector function that represents the curve of intersection of the two surfaces: The cone z = sqrt( x^2 + y^2) and the plane z = 1+y. Homework Equations z = sqrt( x^2 + y^2) and the plane z = 1+y. The Attempt at a Solution This problem can be solved as...
  26. C

    Gradient of an inverse vector function?

    Homework Statement Consider f(\vec{x}) = |\vec{x}|^r, where \vec{x} \in ℝ^n and r \in ℝ. Find \vec{∇}f The Attempt at a Solution I know a vector function maps real numbers to a set of vectors, but here I believe we have the opposite. (inverse of a vector function, assuming inverse...
  27. 6

    Simple vector function problem, find slope of tangent?

    Homework Statement Show that the curve r = (t2,t3-t) Intersects itself at (1,0), and find the slopes of the tangents at this point. Homework Equations The Attempt at a Solution Okay I can show it intersects itself there, but what I am having trouble with is when they say slopes...
  28. N

    Mathematica How to define vector function in Mathematica

    How you define vector function in Mathematica? For example, f is a vector function and f=(xy,yz,zx). How to define this in Mathematica and then how to calculate the value of the components of f for any number x, y, and z? For scalar functions it goes as this: f[x_]:=x^2 f[4] Any...
  29. L

    Derivative of a vector function

    Hi, I am reading through a book called "Matrix Differential Calculus" by Magnus and Neudecker. They go through taking the derivative of a vector in quadratic form that I need help with. For \vec{x} being a vector and A being a constant square matrix \frac {d(\vec{x}^TA\vec{x})}...
  30. Z

    Chain Rule for Vector Function

    Homework Statement I'm trying to figure out how to take grad(f(x(t)) where x(t) is a vector. Since it's part of a physics problem, it's assumed x(t) is in 3-dimensional space. The Attempt at a Solution My guess is that grad(f(x(t)) = ((∂f/∂x)(∂x/∂x),(∂f/∂x)(∂x/∂y),(∂f/∂x)(∂x/∂z)) but...
  31. T

    Line integral with vector function on circular path.

    I'm not getting the answer from the back of the book for some reason. Is the book wrong or am I wrong? Homework Statement calculate \intf · dr for the given vector field f(x, y) and curve C: f(x, y) = (x^2 + y^2) i; C : x = 2 + cos t, y = sin t, 0 ≤ t ≤ 2πHomework Equations itex]\int[/itex]f ·...
  32. B

    How do I determine the y values for the tail of a vector function plot?

    Hi How do I plot this vector function F(x,y) = i + cos x j where i and j are unit vectors Spose I take x = 0, then the components in i and y direction are 1, 1 x = pi/4, then " " 1, 1/sq rt 2 x = pi/2 " "...
  33. D

    How to make a vector function from a f(x) function

    Let's say I want to turn f(x) = x2 into a vector function. How would I do that? I know I can take plots of f(x) = x2 then plug them into the Pythagorean theorem to get the distance from the origin and then I would also know the direction. But is that doesn't seem the same as a vector valued...
  34. B

    Divergenceless vector function - can we draw component by componet conclusion?

    divergenceless vector function - can we draw "component by componet" conclusion? Homework Statement Is this true or false? \nabla \bullet {\bf{A}} = \frac{{\partial {A_i}}}{{\partial {x_i}}} + \frac{{\partial {A_j}}}{{\partial {x_j}}} + \frac{{\partial {A_k}}}{{\partial {x_k}}} = 0{\rm{...
  35. A

    Sketching Vector Function: -yx'+xy

    Homework Statement Sketch a function V= -yx'+xy' ? Homework Equations The Attempt at a Solution i have compared it with r= xi'+yj'. and putting different values of y and x to sketch it on y -axis and x axis. is it correct. how should i do it.
  36. P

    Notation for third order derivative of a vector function

    1. let f: R^n -> R, then f' is a vector and f'' is a matrix, how about f'''? it is a cube? I guess we have to use matrix notation for f'''. I have seen the notation " f'''(x)(h,h,h) ", which is a real number for sure. I have no clue how to operate it though. Any reference on third order...
  37. H

    Arc length of vector function curve

    Homework Statement 1. Find the length of the curve from t=0 to t=1. r(t) = <2t, t^2, (1/3)t^3> 2. Reparametrize the curve with respect to arc length measured from the point where t=0 in the direction of increasing t. r(t) = <e^(2t)cos2t, 2, e^(2t)sin2t>Homework Equations S = \int{r'(t)} dt...
  38. K

    Show that Vector function lies on a sphere

    Homework Statement Let r1 and r2 be differentiable 3-space vector-valued functions. Show that for a differentiable 3-space vector-valued function r, the graph of r lies on a sphere centered at the origin if and only if r(t) and r′(t) are orthogonal (perpendicular) for all t. Homework...
  39. D

    Gradient of a Vector Function in Other Co-ordinate Systems

    Homework Statement I am trying to figure out how to take the gradient of a vector function in polar and spherical co-ordinates. Homework Equations The Attempt at a Solution I am aware of how the gradient of a vector function in cartesian co-ords looks, simply the second order...
  40. X

    Find the derivative of the vector function

    Find the derivative of the vector function r(t) = ta x (b + tc) a=<-2,2,-1> b=<-1,1,1> c=<-2,2,4> I know r(t)=ta x (b + tc)=(axb)t+(axc)t^2 then i got lost
  41. K

    Divergence of a vector function

    Homework Statement Let's define the radial vector \vec{v}(r) = \hat{r}/r^{2} where \vec{r} = \vec{OP} (O being the origin of our coordinate system and P being our observation point at point (x, y, z)). Using spherical coordinates, demonstrate that \vec{\nabla } \cdot\vec{v}(r) = 0 everywhere...
  42. S

    Arc length of vector function with trigonometric components

    Homework Statement Find the length of the path traced out by a particle moving on a curve according to the given equation during the time interval specified in each case. r(t) = (c2/a)cos3t i + (c2/b)sin3t j where i and j are the usual unit vectors, 0 \leq t \leq 2\pi, c2 = a2 - b2, and 0...
  43. Saladsamurai

    Find the Math Forming a Vector Function in 2 Dimensions

    Okay. My reason for posting this is that I need help actually formulating the 'math part' of it. I can get the right answer by 'inspection.' And from the way the book is written, I believe that is how the authors expect you to find it. But for self gratifying reasons, I wish to generalize...
  44. R

    Find some vector function whose image is the intersection of two surfaces

    Hi all, I'm quite new here, but it's been a while since I've been browsing through these forums for past answered questions for calculus and physics, but now comes the time where I'm the one needing help that's not been questioned yet. Homework Statement Find some* vector funcion r with...
  45. M

    How Do I Find the Derivative of This Vector Function?

    Homework Statement Find the derivative of the vector function r(t)=ta X (b+at) where a=<4,5,2>, b=<1,-3,2>, and c=<4,3,1> Homework Equations The Attempt at a Solution I know how to take the derivative and everything but the way this question is worded confuses me! I'm assuming...
  46. Z

    Basic vector function questions

    Homework Statement True or False: a. if k(t)=o, the curve is a straight line b. if the magnitude of r(t)=1 for all t then r'(t) is orthogonalo to r(t) c. different parametrizations of the same curve result in identical tangent vectors at a given point Homework Equations The...
  47. L

    Dot product of vector function?

    Greetings. I was thinking about finding the angle between two functions, so I thought it may be elegant to turn them into vector valued functions, and find the dot product at a given variable value where the vectors lie on the same plane and are functions of the same variable. I'm going to go...
  48. Z

    General true of false questions about vector function in calc 3

    Homework Statement a. the derivative of a vector function is obtained by differentiating each component function b. if r(t) is a differentiable vector function, then d/dt the magnitude of r(t) = the magnitude of r'(t) c. the binormal vector is B(t) =N(t)xT(t) d. if k(t)=0 for all t, the...
  49. 1

    Vector Function (Calc. 3)

    Homework Statement Find a vector function that represents the curve of intersection of the two surfaces: The paraboloid z = 4x^2 + y^2 The parabolic cylinder y = x^2 Homework Equations z = 4x^2 + y^2 y = x^2 The Attempt at a Solution Combining the two equations: z =...
  50. W

    Indefinite integral of vector function

    Homework Statement If {\vec{V}(t) is a vector function of t , find the indefinite integral: \int (\vec{V}\times \frac{d^2\vec{V}}{dt^2}) \,dt Homework Equations The Attempt at a Solution I have solved it by decomposing and integrating each terms of vector \vec{V}\times \frac{d^2t}{dt^2}...
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