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# Vector Calculus

### Mathematics - Vector Calculus

Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t)=(e^2t, e^-2t, te^2t), t=0

### Vector Space Matrix Analysis

Let S be defined as: S = {(x,y): x+2y>=1; x,y=R) Give an example of a vector which is in S Give an example of a vector which is NOT in S Show that S is closed under addition bu NOT under scalar multiplication What can you conclude about S

### Basis of a vector space

Hi. I have the following 2 questions. 1. Find a vector that, together with vectors [1,1,1], and [1,2,1], forms a basis of R^3. 2. Show that the functions (c1 + c2sin^2x + c3cos^2x) form a vector space. Find a basis for it. What is its dimension? Thanks

### The Cross Product of Vectors

Find the area of a parallelogram with vertices K(1,2,3), L(1,3,6), M(3,8,6), and N(3,7,3).

### Orthogonal unit vectors

Find two unit vectors orthogonal to both i + j and i - j + k

### Work done by force field

Please see attached problem set #3 3. Find the work done by the force field F(x,y,z) = zi + xzj + (xy +z)k along a straight line segment from (1, 0, -2) to (4, 6, 2).

### Vectors - Find the Resultant Vector

Make a sketch of the vectors and find the magnitude of the resultant. - Two forces each of 4 units at a 90-degree angle to each other. - Two forces each of 10 units acting at a 120 angle to each other. - Two forces each of 8 units acting at a 60 degree angle to each other.

### Several Vector Problems

Please provide the solution for each section of the problems - Please see the attached file.

### Vectors - Work Done

Let C be the line segment form P(1,0,2) to Q(-2,3,1), and let F be the force given by F(x,y,z)= 2zi - yj + 2xk. Find the work done by F in moving a particle along C.

### Finding a Vector Tangent

Let C be the curve that is parametrically given by R= 3sin(t)i + 4(t)j + 3cos(t)k, 0 is less than of equal to t which is less than or equal to pie. What is the vector T(t) tangent to R(t)?

### Conservative vector fields.

Determine whether or not the given vector field is conservative. If its conservative find f such that F=delta f 1.) F(x,y,z)= yi + xj + k 2.) F(x,y,z)= zxi + xyj + yzk

### Determine Whether or Not F is a Conservative Vector Field

Determine whether or not F is a conservative vector field. If it is, find function f such that F=delta f 1.)F(x,y)= (x^2 + y)i + x^2j 2.)F(x,y)= (ye^xy + (4x^3)y)i + (xe^xy + x^4)j.

### Curl and Divergence

1.)F(x,y,z)= xyj + xyzk 2.)F(x,y,z)= sin(x)i + cos(x)j + z^2k 3.)F(x,y,z)= xe^yi - ze^yj + yln(z)k 4.)F(x,y,x)= e^xyzi + sin(x-y)j - (xy/z)k

### Find the gradient vector field delta f of f and sketch it

F(x,y)= x^2 - (1/2)y^2

### Denote the vector space of all nxn matrices

The material is from ABSTRACT VECTOR SPACE. Please kindly show each step of your solution.

### Vectors and Steinitz replacement

Choose two bases of V3(R) they should have no vectors in common and neither of them should contain multiples of the standard basis vectors e1, e2, e3. a) Prove that they are indeed bases of V3(R) b) Let one of your bases be A and the other B. Illustrate the steps of the Steinitz replacement theorem by converting B into A ste

### Vector Spaces, Subspaces and Dimension

V3(R) represents the set of vectors in 3D space. What kind of geometrical objects are represented by the various subspaces of V3(R)? i.e A 1D subspace S with basis { (0, 1, 0)Transpose} represents the set of vectors parallel to the y-axis, so the set of points with position vectors in s is the y-axis itself. You need only

### Vector Spaces : Set S of vectors is linearly independent

Show that the following set S of vectors is linearly independent in V4(R) 2 1 3 0 -1 -2 1 2 1 -1 2 -1 (The columns should be in brackets and separated by a comma.)

### Trees, Vertices and Edges

Which one of these is correct and why? If T is a tree with m vertices, how many edges does T have? ans: m-1 or If T is a tree with n vertices, how many edges does T have? ans: n(n-1)/2

### Vector Subspace, Orthonormal Basis, Othogonal Projection and Inner Product

Let W be the subspace of R^2 spanned by the vector (3, 4). Using the standard inner product, let E be the orthogonal projection of R^2 onto W. Find (a) a formula for E(x_1, x_2); (b) the matrix of E in the standard ordered basis; (c) W^1; (d) an orthonormal basis in which E is represented by the matrix [1 0 0 0].

### Positive Integer Element of the Vector Space

Let n be a positive integer. Let A be an element of the vector space Mat(n,n,F), which has dimension n2 over F. Show that the span of the infinite set of matrices span(In, A, A2, A3, ...) has dimension not exceeding n over F. Defn of the linear space Mat(n,n,F): The set of all n-by-n matrices with entries in F. Mat(n,n,F )

### Vector Magnitude Proof

Show |u+v|^2+|u-v|^2=2|u|+2|v|^2

### Unit Vector and Magnitude

A)Find a unit vector in the direction of u=<3,2,5> b) Find magnitude U for u=a+b where a =-3i+j & b=3i-5j

### Lagrangian Strain Tensors and Differential Vectors

Please see the attached file regarding specifics. Thank you so much for your help.

### Normed Vector Space

Consider the vector space R^2 with the norm &#9553;(x,y)&#9553; = &#9474;x &#9474;+&#9474;y &#9474; Show that the set U = { u element of R^2 : 0< &#9553;u&#9553; < 1} is an open set in this normed vector space.

### Vector Subspace Proofing

Show that Cn[a,b] is a subspace of C[a,b].

### Shortest path between two points on a plane.

Let A = (0, 1) and B = (3, 2) be points on a plane. What is the length of the shortest path from A to the x-axis to B? Find where the path should touch the x-axis for this minimum to be attained and argue why it is the minimum.

### Vector Spaces: Direct Tensor Notation

Problem D: Using direct tensor notation (ie., without resorting to component forms) prove that if u, v, w are real numbers f V_3 and L, B are real numbers of R a) (L u B v) = L B (u v) b) (u (v + w)) = (u v) + (u w) Hint: Take each expression and operate on a vector alpha is a real number of V_3 and proceed accordingly

### Solving: Multidimensional Arrays and Vectors

I need the following in C++: A certain professor has a file containing a table of student grades, where the first line of the file contains the number of students and the number of scores in the table; each row of the table represents the exam scores of a given student and each column represents the scores on a given exam. Th

### Multidimensional Arrays and Vectors

I need the following in C++. The output needs to be in a table format similar to the following sample: A demographic study of the metropolitan area around Dogpatch divided it into three regions (urban, suburban, and exurban) and published the following table showing the annual migration from one region to another (the number