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# Vector Calculus

### Vectors in spherical and cylindrical

(a) Given A = a*p_hat + b*psi_hat + c*z_hat (cylindrical unit vectors), where a, b, and c are constants. Is A a constant vector (uniform vector field)? If not, find: the divergence and curl of A (b) If A = a*r_hat + b*theta_hat + c*phi_hat in spherical coordinates, with constant coefficients. Is A a constant vector (unifor

### Hyperbola : Vertices and Foci

Find the coordinates of the vertices and the foci of the hyperbola and sketch the graph. 9y^2/25 - x^2 = 1

### Ellipses : Vertices, Foci and Eccentricity

1) 9x^2+4y^2+36x-24y+36=0 The answer is as following: Vertices: (-2,6), (-2,0) Foci: (-2, 3+or -&#8730;5) Eccentricity: &#8730;5/3 I have listed the answers for no. 1 above, but don't understand the steps. Please explain. 2) x^2+5y^2-8x-30y-39=0 Center: (4, 3) Vertices: (-6, 3), (14, 3) Foci: (4 + or -

### Graphing : Writing Polynomials from Word Problems and Where should a pilot start descent?

An approach path for an aircraft landing is shown in the figure and satisfies the following conditions: (i) The cruising altitude is h when descent starts at a horizontal distance e from touchdown at the origin. (ii) The pilot must maintain a constant horizontal speed v throughout descent. (iii) The absolute value of the ver

### Vector Spaces

How to prove or counter with example the following statements: (1) If two subspaces are orthogonal, then they are independent. (2) If two subspaces are independent, then they are orthogonal. I know that a vector v element of V is orthogonal to a subspace W element V if v is orthogonal to every w element W. Two subspaces W1

### Vector calculus: Flux and Surface Integral

#2) Flux integrals; &#61682;s&#61682; F?n dA. Evaluate the integral for the following data. Indicate the kind of surface. (show the details of your work): F=[x2, ey,1], S: x+y+z=1, x&#61619; 0, y&#61619; 0, z&#61619; 0 Kreyszig's Advanced engineering mathematics 8th ed.: section 9.6: Surface integrals Please

### Partial Order, Linear Functional, Vector Space and Subspace

Let be a vector space and a subset of such that implies and for Define a partial order on by defining to mean . A linear functional on is said to be positive (with respect to ) if for . Let be any subspace of with the property that for each there is an with . Assume that , where Then each positi

### Proof of Vertex, Extreme Point, Basic Feasible Solution

Can you please let me know how to approach those proof questions. Consider the polyhedron P = {x &#61646; Rn : xi > 0 for all i = 1 ... n}. a)Prove that the origin (i.e. the vector of all 0's) is a vertex of P, according to the definition of a vertex (i.e. do not rely on the fact that vertex = extreme point = basic feasibl

### Area Measure and Orthogonal Vectors

Let m be an area measure on {z in C:|z| < 1}. Show that 1, z, z^2,... are orthogonal vectors in L^2(m). Find ||z^n||, n >= 0. If e_n=(z^n)/||z^n||, n >= 0, is {e_0, e_1,...} a basis for L^2(m)?

### Find lengths of a square given opposite vertices.

Two opposite vertices of a square are P(3, 2) and Q(-5, -10). Find the length of: a) a diagonal of the sqaure b) a side of the sqaure

### Position vector

The vector v has an initial point p of (7,3) and terminal point q of (-5,-4). How do I find its position vector and write v in the form of ai + bj?

### Subtraction of vectors

Given these vectors how do I find u - v? U=-8i-6j V=-10i+5j

### Vectors

Using the given vectors how do I find the specified dot product u=3i-8j;v=4i+9j find u.v

### Vector Integrals

- In space, given 3 points M (1,0,0), P (0,2,0), Q (0,0,3). Let C be the contour... (See attached file for full problem description).

### Vectors, Dot Products and Orthogonality

3. Two vectors Xand Y are said to be orthogonal (perpendicular) if the angle between them is r/2. (a) Prove that X and V are orthogonal if and only if X . Y = 0. Use part (a) to determine if the following vectors are orthogonal. (b) X =(?6,4.2) and Y =(6,5,8) (c) X=(?4,8,3) and Y=(2,5,16) (d) X = (?5. 7, 2) and Y = (4, 1, 6

Important Formulas and their Explanations (III): Gradient, Divergence and Curl Gradient of a constant Gradient of a con

See attached

### Dot Product and Angle

If v = i - j and w = i + j, find the dot product v * w and the cosine of the angle between v and w.

### Vectors

If v=3i-4j, find ||v||

### Vector Field, Gradients, Div, Divergence, Curl and Surface Integrals

4) A vector field F is shown. Use the interpretation of divergence derived in this section to determine whether div F is positive or negative at P1 and P2 a.) Are the points P1 and P2 so sources or sinks vector field F shown in the figure? Give an explanation based solely on the picture. 5)Use the divergence Theorem to c

### Potential and Electric Field Vector of Two Concentric Charged Cylinders

B. A region is surrounded by two infinitely long concentric cylinders of radii, a1 and a2 (a2>a1). The concentric cylinders are charged to potentials phi1 and phi2 respectively. Determine the potential and electric field vector everywhere in the region. Please see the attached file for the fully formatted problem.

### Potential Functions: Conservative Vector Fields

For each of the following vector fields F type in a potential function f (please see attached for full question). See the attached file.

### Vector Calculus : Path Independence of Central Force

Show ... is independent of path for any central force F....

### Vectors : Find the i-Component of a Force

Consider the force F shown below... [DIAGRAM] Select the option giving the i-component of F.

### Vector Calculus Concepts

See the attached problem.

### Vector Calculus: Irrotational Function

Please see the attached files for the fully formatted problems.

### Vector Calculus : Flux and Gauss's Law

(a) Consider a vector function with the properly ... = 0 everywhere on two closed surfaces S1 und S2 and in the volume V enclosed by them (see the figure). Show that the flux ol F through S1, equals the flux of F through S2. In calculating the fluxes, choose the direction of the normals as indicated by the arrows in the figure.

### Vector Calculus - Flux and Magnetic Fields

(a) One of Maxwell's equations states that V H = 0, where 11 is any magnetic field. Show that if...for any closed surface S. (b) Determine the flux of a uniform magnetic field B throngh the curved surface of a right circular cone (radius R, height h) oriented so that B is normall to the base of the cone as shown in the figure.

### One Dimensional Normed Linear Space : Completeness and Continuity

Suppose that E is a one-dimensional normed linear space. Prove that E is complete and that each linear functional on E is continous.

### Normed Linear Space : Continuity

Suppose that N is a normed linear space. Prove that every linear functional on N is continuous if and only if N is finite dimensional.