Suppose d is Show that p: X?> [0,00) defined by a metric on X, p Cxy)= 3d (x,y) %3D 2+3d(x, u) is also a metrie on X

Q: „Consider the space (R,T). where t={R,Ø}U{ACR:[0,1)CA} :Then (R,T) is not T, O T, but not T, O T2 O…

A:

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: . Let T be the triangular lamina in Figure 21. O Show that the horizontal cut at height y has length…

A:

Q: Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition…

A:

Q: Let d be metric on X, and de f ined di on X as di = max (d (r, ).0}, then dis semi metric but not…

A: Given d be an metric on X and d1 is defined on X by  d1=maxdx,y,0                     (i) To verify…

Q: 7.H Jse the divergence theorem to evaluate where S is the sphere x+y? +? = 4 oriented with outward…

A:

Q: Evaluate both sides of divergence theorem for A = pa,+pcos(@)a , + za , through the closed surface…

A: Given vector field is  Divergence theorem

Q: Suppose F(x, y) = (3x – 2y)i + 3xj and C is the counter-clockwise oriented sector of a circle…

A: Given, F→(x,y)=(3x-2y)i→+3xj→ Here, M=3x-2y and N=3x radius=3 θ=π3 Limits are 0≤r≤3 , 0≤θ≤π3

Q: Evaluate : //| cos(x² +y³) dV, where E is the region bounded by the disk x² + y? < 9 E and the plane…

A: From the region of integration which is bounded by x2+y2≤9 and z=1, identify the limits of x, y and…

Q: Assume s is a smooth surpace parametrized by e (u,v): <Jv, win' u, SZ-v ). Evaluatk 2x sin y do.

A:

Q: 21. Show that the volume V of a region D in space enclosed by the oriented surface S with outward…

A:

Q: Use the Divergence Theorem to evaluate F. ds, where F(x, y, z) = z?xi + (+ tan zj + (x²z + y2)k and…

A: We are given a closed surface integral and we have to use the Divergence Theorem to evaluate that.

Q: Write a homeomorphism between the unit closed disk A = {(r,y E R2 : a2 + y? < 1)} and the unit…

A:

Q: Let X=R2 and let d be the Euclidean metric.  Define d3:  R2 x R2 to R by d3(x, y) = min{1, d(x, y)}…

A:

Q: Consider the following mappings on RXR: d1(x,y) = e*-Y, d2(x,y) = |2x –yl and d3(x,y) V=yl O Both d1…

A:

Q: Let d: R x R R. defined as d (z, y) = |#1 - ml wherez= (21,2), y= (y.), then d is %3D Osemi metric…

A:

Q: b.) show that brnormal of the muolute the unit *-ズ+(c-s)で g =dG) |k-s) K| K* where K is the…

A:

Q: Exercise. a) For every ,r, y E R, let p(r, y) =| x² – y? |. Is pa metric U..... on R? b) For every…

A:

Q: If F = xy?i + yz? j + zx2k, and S is the region bounded by the triangle with vertices (1,0,0),…

A:

Q: Show that if u is a unit vector, then the arc length parameter along the line r(t) = P0 + t u from…

A: It is known that,

Q: Let F(z, y, 2) = 12?ri + (ty + tan(z))j + (1z?z+ 4y?)k. Use the Divergence Theorem to evaluate f, F…

A: We must apply the Divergence theorem on a closed solid, but the surface S is not closed (it is just…

Q: Exercise. a) For every ,x, y E R, let p(x, y) =| x? – y? |. Is p a metric on R? b) For every r, y E…

A:

Q: Suppose F(x,y)=(4x−4y)i +3xj  and C is the counter-clockwise oriented sector of a circle centered…

A:

Q: Let (x, y) and (z, w) be two points in R? state which of the following does not define metric on R?…

A:

Q: At what points (x,y.z) in space are the tunctions continuous? a. h(x.y.z) = In (4z² - 4x – 2y³ - 5)…

A: This is a problem of multi variable calculus.

Q: Let X = R and x, y E R. Admit for all x and y the metric d(x, y) = |arctan x arctan y|. | Compare…

A:

Q: -) Let -y M(x, y) = x2 and N(x, y) = + y? x2 + y? how that ƏN ƏM dx ду ) Show that -y F(x, y) = x² +…

A: Given :-  A vector field for conservative field in the plane ,  ∂N∂x = ∂M∂y To Determine :-  M x,y =…

Q: Let C be the portion of the circle x² + y² = 4 from (2,0) to (0,2) traced counterclockwise. (a) (b)…

A: As per the question we are given a curve C which is a quarter circle defined as : x2 + y2 = 4  ,…

Q: Compute the contour integral dz, where l'is the contour of the perimeter of the square with vertices…

A: We will do it  by dividing r into 4 paths.

Q: Let f(x, y, z) = In (x² + y + z?). The level surface f (x, y, z) = In 4 is a: Elliptical cone around…

A: Given  f (x,y,z) = ln( x2+ y+z2),To find : level surface f= ln4

Q: Find the volume of the Intersection curve C C Projection of C on ay-plane

A: Given: Given paraboloids are z=x2+y2 and z=8-x2-y2 We want to find the volume bounded by the…

Q: evaluate.both sides of divergence theorem for A= ra,+rcos(e)a, +ra, through the closed surface…

A:

Q: Let d(x,y) = |1/x - 1/yl be a metric on RI(0}XRI(O}. Then the open ball of center 1/2 and radius 1/3…

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

Q: Question 4. Prove that the metrics defined on R? with the following rules; 1/2 de (x, y) = E (Tk –…

A:

Q: Let W = Span{u} where u -2. Let y = |1, find the distance from y to W L3]

A:

Q: If R is a closed bounded region in the xy-plane whose boundary curve is C with arc lengths and…

A: i. Consider that F be a vector field and F→=Mi+Nj and C is a simple closed curves positively…

Q: If F=xi+yj, prove that JF.ñds equals [V.FdA, where R is the disk of radius C R "a" and centre at the…

A: Given Data: F =xi  +yjTo Prove:∫C F . n ds =∫R∇·F dA where R is the disk of radius ''a'' and centre…

Q: Let d : R x R → R be defined to be d(x, y) = | arctan(x) – arctan(y)|. Show that d is a metric on R.

A:

Q: „Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} f A=[0,1], then the interior of A is (0,1) O…

A:

Q: Let E denote the part of the plane x + y + z = 1 that is inside the cylinder r? + y? = 1. Compute…

A: We have to find the value of  ∬ F→.dS→

Q: Verify that (S, d) is a metric space. 10.2 Prove that d(x, y) < d'(x, y)s/2 d(x, y) for all points'…

A: To prove dx,y≤d'x,y≤2dx,y for all points x,y∈ℝ×ℝ where d is the usual metric and d' is the…

Q: Suppose F(x, y) = (4x – 3y)i + 4xj and C is the counter-clockwise oriented sector of a circle…

A:

Q: Use the Divergence Theorem to evaluate F. dS where S is the sphere r² + y² + z² = 9 and F = (3r, 2y,…

A:

Q: „Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} :lf A=(0,1), then the closure of A is (0,1)…

A:

Q: Let d(x.y) = |1/x - 1/y be a metric on R(0)XR\(0). Then the open ball of center 1 and radius 1/4 is*…

A:

Q: Let X = R², x, y EX where x = (x₁, x2), y = (y₁.Y2) with the metric function (4 pts) (x, y) =…

A: This is a question of topology and metric space.

Q: Evaluate both sides of divergence theorem for A= pa, + pcos(0)a , + zaz through the closed surface…

A:

Q: Let F(x, y, 2) = 32?ai+ (y³ + tan(z))j+ (3x²z- 4y?)k. Use the Divergence Theorem to evaluate fs F dS…

A:

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.