1. Given the map f:R4R³ defined by f(x,y,z,w) = (0,0,0) then the dimension of Kerf is:A. 2B. 0OC. 4D. 3O E. 1

Dimension of Rn is n. Let T:V→W be a transformation. Kernal of a transformation T is defined as,…

Answered: 1. Given the map ƒ:Rª→→→→R³ defined by…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. Let X = R². For any (x₁, y₁), (x2, y2) € R², define d₂((x1, y₁), (x2, Y2)) := √√(x2 − X1)² + (Y2 — Y1)². Show that d₂ is a metric on R².
3. Let H = 2xi – 3xy j+ xz² k. Calculate fs H · dÃ, where S is the surface of the cube with corners at (0,0,0), (1,0,0), (0,1,0), (1, 1, 0), (0,0,1), (1,0,1), (0,1, 1), and (1, 1, 1), oriented outward.
Provide the General Solution of each differential equations. Box your Final Answer and show your solutions:
Please do fast and correctly
A. Let f(x, y, z) = z² - x² - y² .The set of all values of c such that the level surface f(x, y, z) = c²-2c-3 intersects x - y plane in a circle 1- (-∞, -1) U (3,00) 2- (1,3) 3-(-1,3) 4-(1,-3). ling
If f= (y? -z? +3yz – 2x)i+(3xz +2.xy)J+(3xy – 2.xz +2z)k then | which of the following is/are true? A) f is solenoidal B) f is an irrotational c) ƒ is both irrotational and solenoidal D) f is an irrotational but not solenoidal
Consider the following curves C : x³ + y³ C2 : x³ + y³ = 9 = 1 C3 : y = x +1 C4 : y = 0 that are shown in the figure below. R Q C4 -1 C2 C3 -1 C1 Define the following regions, also shown in the figure above. Q is the region bounded by C3 on the left, C1 on the right, and C4 below. R is the region enclosed by all four curves: above C, and C4, below C, and C2. (a) Use a Riemann Sum to approximate the area of the region Q. Use any rule we studied in class, and n = 4 subintervals. (b) Set up integral(s) to compute the area of the region R. (c) Set up integral(s) to compute the volume of the solid obtained by rotating Q around C4.
2: Let a function d: R2 x R? →R defined by d(x, y) = ((x1 – y1)² + (x2 - y2)?) Vx = (x1, x2), y = (y1, Y2) E R2, then d be a %3D metric function on R2.
3. For the map F : R³ → R² defined by F(x, y, 2) = (x²y + e², yz – x) Calculate F' (x, y, 2) • If p = (1, 1,0) and vp = (1,0, 0), what is the push-forward F.(vp)? %3D
) Let f(z) = (z − ₁)(z − 3₂)... (zn), where ₁, 2,...,n E C. If y is a positively oriented simple close contour closing every ; in its interior, prove that $ dz ƒ(z) = 0.
Exercise 4.1. Let F : R² → R² be the map defined by F (G) Describe the image by F of the points lying on the unit circle centered at 0, i.e. { (4) e R² | x² + y² = 1}. 3y) for any (G) E R².
1. Given the map f:R4R³ defined by f(x,y,z,w) = (0,0,0) then the dimension of Kerf is: A. 1 B. 3 C. 4 D. 2 OE. O

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1. Given the map ƒ:Rª→→→→R³ defined by f(x,y,z,w) = (0,0,0) then the dimension of Kerf is: OA 2 O B.0 OC.4 OD. 3 OE. 1