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.
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...
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...
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##?
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...
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...
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...
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...
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...
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...
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?
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...
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...
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?
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!
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...
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...
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?
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)...
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...
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...
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...
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...
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...
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...
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...
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...
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})}...
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...
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 ·...
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 " "...
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...
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{...
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.
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...
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...
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...
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...
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
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...
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...
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...
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...
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...
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...
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...
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...
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 =...
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}...