12. Show that d((x1, Y1), (x2, Y2)) = |x1x2- Y1Y2| is not a metric on X = R %3D %3D

Q: Show that d given by d(x, y) = |x-ylis a metric on R by using definition 4.2.1. You may want to use…

A: we knew  that given metric d(x,y) = |x-y| is always ≥0.  now d(x,y) = 0 ⇔ |x-y| = 0…

Q: Evaluate the line integral [6. 6ayⓇ ds, where C is the right half of the circle x² + y² = 4.

A: To evaluate :   ∫C6xy6ds, where C is the right half of the circle x2+y2=4 Prerequisite : We can…

Q: (15) Let D be the half-disk x² + y? 0. Compute / | (a + y) dA D

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Q: (a) Prove that M: (x² + y^y° + 3z² = 1 is a surface. (b) For which values of e is M: z(z – 2) + xy =…

A: Consider the provided question, (a)

Q: 20. Show that a²F(x, y) -dx dy дх ду over the rectangle xo < x < x1, Yo < y < y1, is F(x, у) — F(x0.…

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Q: If F= 2y I- 3J +x²k and s the surface parabolic cylinder y= 8x in bounded by the planes OF the 2.…

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Q: 13-Evaluate JSS zdV E where E lies between the spheres x2 + y? + z² = 1and %3D x? + y? + z? = 4 in…

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Q: If R is the region boundeb by the square a|+|y = 1 then ffR(y-2)dA is equal

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Q: 3) Prove thet the xy-plane with constant curva tue k= urve In is o circle of radious

A: We have to prove that a curve in the xy plane with constant curvature k=1r is a circle of radius r

Q: The area bounded by the curve y3a and the lines z 4 and a-9 above the z-azis is aquare units O 27v0…

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Q: Q2: (a) Let M = {x,y,z) and define d: MXM → R by Determine whether d is a metric on M. d(x,x) = d(y,…

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Q: 3. Find the area of the band cut from the paraboloid x² + y² -z = 0 by the planes z = 2 and z = 6.…

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Q: |Use Green's Theorem to evaluate | -x'ydx+xy°dy where C is the boundary of the C circle x + y =16…

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Q: Find the rates at which the volume, surface area, and interior diagonal are changing at the instant…

A: Volume of rectangular box (V) = xyz Take derivative with respect to t, we get…

Q: Consider 2-dimensional hyperbolic space with metric ds ds'+dy and y > 0. The length of a line…

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Q: 4 Find the curvature k of the plane curve y = x +- at x = 1. K =

A: Given curve is , y= x + 4x  at x=1

Q: Let d: Rx R*→[0, ∞) such that d(XF)=√(x, y)² show that (R.d) metric space.

A: `

Q: Compute where Cis the counter clockwise oriented boundary of upper-half unit disc D. 0.66666666667…

A: We need to find ,  ∮ y2dx + 3xydy , over  C ,  where C is the counter clockwise oriented boundary of…

Q: Suppose D is the region in the plane bounded by the lines x + y = 0, x +y = 3, x – 2y = 0, and x –…

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Q: 38. Calculate y = x, x = 0, and x = 1. || e* dA, where D is bounded by the lines y =x +1,

A: We have to find ∫∫D ex dA, where D is bounded by the lines y=x+1, y=x, x=0 and x=1

Q: If C is the line segment from (1,3) to, (0, 0), find the value of the line integral: S(3y² i +3x}) ·…

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Q: (b) ſf f(x,y) dA if f(x, y) = y², where R is the triangular region with vertices (0, 1), (1, 2) and…

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Q: 1)calculate a*yz dædydz where D is the rectangular parallelepiped 1< « < 2,1 < y < 2,1 < z < 5

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Q: A surface x(y, z) is defined implicitly xy + yz +x'z' = 3-xZ Calculate ду

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Q: Let X=R2 and defined d2:  R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a…

A: Given a function d2: R2×R2 → R, defined by d2x1,y1,x2,y2=maxx1-x2,y1-y2. To verify that d2 is a…

Q: å:RR→R be defined by dcx.y)=|xl+1yl, HxixR (R,d)is Than a metric space.

A: As per our guidelines we supposed to solve the very first question when the multiple questions are…

Q: # 4. Evaluate the line integral | (x +2x²y) ds, where C is the upper half of the unit circle x2 + y2…

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Q: 4. Evaluate the line integral / r y* ds where C is the right half of the circle r² + y² = 16.

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Q: Let P be the plane 2x + y – z = 2 and let Q(1,0,3) be a point. We want to find the minimum distance…

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Q: || x dA, R is bounded by y = x, 2x + y = 6 and the y-axis. R

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Q: Find minimum perpendecular distance between given paraboles. y = x2 + 1 y = x - x2

A: To find the minimum distance between 2 parabolas

Q: 4 6 Let y = 7 2 40 Compute the distance d from y to the plane in IR spanned by ui and uz d =

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Q: 2.7 Describe cll soilutions of Asso where A is the Yow ta the matirix meg Vse Y S, and/or t as in…

A: Given  A=1-6-50-2-100100-200001-7000000 Here, we have to solve Ax=0

Q: 2. Calculate (y+ z)dx+(x+z)dy + (x+ y)dz , where C consists of line segments from (0,0,0) to (3,2,4)…

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Q: 3. Evaluate ox?y dx + xy? dy using Green's y 2 Theorem, where C is positively oriented by line y =x…

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Q: Consider 2-dimensional hyperbolic space with metric ds = dr'+dy and y > 0. The length of a line…

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Q: 4. Evaluate ||| x y dV , where E is bounded by the parabolic cylinders y = x², y' = x and the planes…

A: The region E is bounded by the parabolic cylinders y = x2, x = y2 and the planes z = 0 and z = x +…

Q: Calculate (y+z)dx+ (x+ z)dy+ (x+y)dz , where C consists of line segments from (0,0,0) to (3,2,4) and…

A: We will use parametric equation of lines to find the integral

Q: If C is the line segment from (1, 4, 2) to (3, 2, 1), evaluate the line integral y+ 2) ds.

A: Express the line segment in its parametric form and then use the formula below to evaluate the line…

Q: 4. Suppose that VxF= (0,1,-1), S is the plane x + y + z = 1 in the first octant and Cis the boundary…

A: Given: ∇×F=01-1 S is the plane x+y+z=1 in the first octane  and C is the boundary of S. Now, to find…

Q: Evaluate (x + 5y) dV where E is bounded by the parabolic cylinder y = x² and the planes z = 3x, y =…

A: given is  a partial differential equation in dX , dY, dZ we can partially integrate the given…

Q: 5) Consider the surfaces S1 : ryz = 10 and S2 : z =r² + y and the %3D point Po = (1,2,5).

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Q: Findd Fhie boundled by Ahe Cureo area 2. X= 3J-y' and x+Y=3 ーレ

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Q: Evaluate the line integral Levis where C is the half circle x² + y² = 1, y ≥ 0. x² + y² ds

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Q: 2. Suppose that the edge lengths r, y, and z of a closed rectangular box are changing at the…

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Q: Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following…

A: To determine the rate at which the box’s (a) Volume, (b) Surface area and (c) diagonal length are…

Q: se the Divergence Theorem to evaluate (8x + 12y + z2) ds Is here S is the sphere x2 + y2 + z² = 1.

A: To compute the below integral using divergence theorem.

Q: Evaluate dV, where E lines between the spheres a+ y + z = 9 and E 2 + y2 + z? x2 + y? + z2 = 16 in…

A: The answer is in step 2

Q: 3xy dV, where E is bounded by the parabolic cylinders y = x² and x = y and the planes z = 0 and z =…

A: It is given that, y=x2 and x=y2 and the planes z=0 and z=7x+y. Define the region E as follows.…