(b) Is (R2,d) a metric space, where d((x.y1).(x2.y2)) 1x1-x2l; for all (x, , y1 ), (x2 2) E R2 ? Explain your answer.

Q: - || e dA, where D is the open disk D = {(x, y) : x² + y² < 4}.

A:

Q: 7. D is bounded by the surface (r2 + y² + z²)² = a²(x² + y² – 22). %3D

A: We have to find volume where D is bounded by the surface (x² + y² + z²)² = a²(x² + y² — z²).

Q: Calculate , z² dS if S is the prism whose base is the triangle in the limited xy plane by the x axis…

A: Here given figure is Since prism has 5 surfaces (1)-OAB (2) OADE (3) CDE (4) OBCD (5) ABCE

Q: Find the point closest to origin in R³ that is on both the planes 3x+y+z=5 and x+y+z=1.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Consider the following mappings on RXR: * d1(x,y) = ex-y, d2(x,y) = |2x -y and d3(x,y) = √√x - yl O…

A: The given problem is to check which mappings are metric in R. We have to check metric properties of…

Q: Please solve this problem

A: Given integral is,

Q: use green's theorem to evaluate ∫c 5y^3dx-5x^3dy where C is the circle x^2+y^2=4 counterclockwise

A: Given: ∫c 5y^3dx-5x^3dy where C is the circle x^2+y^2=4

Q: Determine whether the space curve given by r(t) = (12,12 – 21 – 3,1 – 3) intersects the x-axis, and…

A:

Q: Bounded by the coordinate planes and the plane 3x + 2y + z = 6

A:

Q: Find the distance between the planes 5x - 3y + z = 9 and 5x 3y + z = -2.

A:

Q: Use Divergence Theorem to evaluate .(v²z²i +z²x²j+x²y²k).ds, where S is the upper part of the sphere…

A: Gauss Divergence Theorem : ∫∫SF·ndS=∫∫∫VdivF dV From ∫∫Sy2z2i+z2x2j+x2y2k·dS, F=y2z2i+z2x2j+x2y2k

Q: 22 Prove that the line Z=2+46 is perpendicular to thme plane X-2y+2=6

A: As per our guideline we are supposed to do 1 question at a time. Please post other questions next…

Q: A(n)________ is the collection of points in the plane the difference of whose distances from two…

A: The given statement is "A(n)________ is the collection of points in the plane the difference of…

Q: Verify Green's theorem in the plane for M = -2y and N = x , where C is the semi circle x2 + y? = 9,…

A:

Q: A. Show that the planes (x+y- z = 1) and ( 2x - 3y + 4z = 5 ) are neither parallel nor…

A: Let's find.

Q: Find the point on the plane 3x + 4y +z = 8 that is nearest to (2,0,1).

A: Given the equation of plane:

Q: What is the projection of r(t) = ti + t4j + etk onto the xz-plane?

A: Given: r(t)= ti+t4j+etk

Q: Evaluate // x² zdS, where S is the part of the S 1 + 4x + 3y above the rectangle plane z = [о, 2] x…

A:

Q: I, Use Stoke's theorem to evalueate S.Fidr, where F Cau2)= - it +x3 + 2î aind C is the Ceurve of…

A:

Q: 2. Let d1, d2 be two metrics on X + 6. Show that: d(r, y) = d1(x, y) + d2(x, y).

A: I have proved all four conditions of metric for d

Q: Find the point on the plane x+ 2y - z 12 that is nearest the origin.

A:

Q: (a) Between 1 and the disk x²+ (1+x² +y*) (b) Between % = 1 and the entire xy-plane. (1+x* + y?)*

A:

Q: Find all planes in R3 whose intersection with the xy-plane is the liner( t) = t ( 2, 1, 0) .

A: GIVEN  Intersection with the xy-plane is the line r( t) = t ( 2, 1, 0)

Q: 2. Do the three planes x + 2y + z = 4, y - z = 1, x+ 3y= 0 have at least one commom point of…

A:

Q: Use the Divergence Theorem to evaluate (6x + 7y + z2) dS where S is the sphere x2 + y2 + z? = 1.

A:

Q: Use Divergence Theorem to evaluate [0²z?i+z?x²j+x*y²k).ds, where S is the upper part of the sphere…

A: To evaluate the below integral using the divergence theorem. ∫∫Sy2z2i+z2x2j+x2y2kds Here S is the…

Q: Find the point on the plane 2x + 3y + z = 7 that is nearest to​ (2,0,1).

A: NOTE: Refresh your page if you can't see any equations. . the given point is . the given plane is…

Q: Evaluate Sf,(z+ 3y – x²) dS where S is the portion of z = 2 – 3y + x² that lies over the triangle in…

A: Ans :

Q: Which among the following diagrams correspond to the metric below on the real plane R^2?…

A: Given diagrams are, Given metric is dp1,p2=sup x1-x2, y1-y2, where p1=x1, y1 and p2=x2,y2 Find the…

Q: or tho gonal Use the least, me thad to find the squones projetiom of bz Ġ 2 -a anto the salwmin of…

A: Consider the given:         A=011211and  b=12-2T=12-2

Q: Consider the following mappings on RXR: * d1(x, y) = e2z-y, d2 = |2x – 2y|, and d3 = VI2x – y|. %3D…

A:

Q: Consider the following mappings on RXR: * d1(1, y) = e2"-y, d2 = |2x – 2y|, d3 = V[2x - yl- and |…

A:

Q: A. Show that the planes (x+ y– z = 1 ) and (2x – 3y+ 4z = 5 ) are neither parallel nor…

A: let the normal vectors for the vectors be a1, a2 thus a1=&lt;1,1,-1&gt; and a2=&lt;2,-3,4&gt;. Let…

Q: 2. Do the three planes X + 2y + z = 4, y - z = 1, x+ 3y = 0 have at least one commom point of…

A: The equation of the plane is given by the formula ax+by+cz+d=0. The system of planes can be solved…

Q: dA, where K is the disk r + (y 2 +?

A:

Q: JJs where S is the sphere x² + y2 + z2 = 1.

A: Given S is the sphere                         x2+y2+z2=1 To evaluate:             ∫∫S3x+3y+z2dS…

Q: find the line of intersection of th e given planes. 3x + 2y + z = -1 and 2x - y + 4z = 5

A:

Q: 9. Suppose that a : R³ → R³ by perpendicular projection onto the plane 3x – 2y + z = 0. Find ↑

A:

Q: Evaluate ([(3xi +2yi)-d$ -d$ where S is a sphere x +y +z? =9.

A: To evaluate the below integral. ∫∫S3xi+2yj⋅ds Here S is a sphere x2+y2+z2=9

Q: find the line of intersection of th e given planes. 4x+ y + z = 0 and 2x - y + 3z = 2

A:

Q: An _____ is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and…

A: The sum of the distances to any point on the ellipse (x,y) from the two foci (c,0) and (-c,0) is a…

Q: +4z=8 and

A: The equation of the plane as, x+z=1orx+0y+z=1 Here, a1 =1,b1 =0,c1 =1. and x+2y+4z=8 Here, a2 =1,b2…

Q: Find b such that B = (0, b) has hyperbolic distance of u from the point A (0,0).

A: Given: B=(0,b) has  hyperbolic distance of u from the point A=(0,0) Formula used:

Q: Let C be the boundary curve counterclockwise of the semidisk æ? + y? 0. Then fc(-2ydx + 3xdy) 5dydx…

A:

Q: Consider the case D+ ={(1,0),(0,1)} = {x1,x2} = {e1,e2} D− ={(−1,−1)} = {x3} = {−e1 −e2} D=D−∪D+.…

A: Please solve. MUST  Verify your solution using R or Python. Please include code and printout as part…

Q: Find the null space associated with the equations: 2x+z+w = 0 7x-6y–z–w-v = 0 -5x+4y+w = 0

A:

Q: 2. For points p = (z1,Yı) and q = (12, 42) € R², the taxicab metric is given by d(p, q) = |r2 - 1|+…

A:

Q: Find the point of intersection between the planes TT :x+y-z =1, n, : 2x – 3y -9z = 10 and n, : x +…

A: The three planes are, π1=x+y-z=1 π2=2x-3y-9z=10 π3=x+2y-3z=-4

Q: Given F(x.y.z) (x+y+Zi +(x-yj+(xz-2)k what is CurlF at the point (1,-1,3)? You should write your…

A: