Evaluate |/I. ze*+ydV where B is the box determined by B 0

Q: 4:: what the Null Space of is T;: R3 R³( T,x= xx a) R, R (T = X.Q

A: As per guidelines, we will solve first three sub-question: We have to find the null space of  1.…

Q: 2. An is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and two…

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

Q: SEr up AN INTEGNAL KEPRESENTAT7 ON OF THE ARGA OFTHE REFGrON BOUNDEO BY y =x3,y=0, x=-l, X= 2 ド

A:

Q: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

A:

Q: 1- iz If z = x +iy and w =- z-i show that w = 1= zis purely real

A: Use modulus property

Q: Evaluate | Vtan a sec" zdx. O a. 2 z tani z +tani +C 2 O b. 2 tani æ + tan +C 2 Ос. 2 O c. 2 2 t…

A: To evaluate the integral,  ∫tanxsec4xdx Rewrite the term sec4x=sec2x·sec2x…

Q: 2. Let X, Y be nonempty sets of real numbers that are bounded above. Let X + Y = {x + y:x € X and y…

A:

Q: 21. Let A and B be two on-empty bounded sets of positive embers and let С%3D {ху: х€ А and y € B}.…

A:

Q: f(x,y) = (In(y + x²)) x² + y² D = {(x,y) ERIy2x²-1 and y ≤4-e-*) (b) Observe the set Is D…

A: (b) The given set is D=x,y∈ℝ : y≥x2-1 and y≤4-e-x To find: The given set is bounded or closed or…

Q: Evaluate / (x - 2y + z) ds. S: z = 3 - x, 0sxs 3, 0sys 2

A:

Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann eguations if u, = v, and u,…

A: Thje given functions are,u(x,y)=y(x2+y2) , v(x,y)=x(x2+y2)

Q: De'Morgan Theorem to prove that: (x.y'+x'.y)' = x.y+x’.y' %3D wwww

A:

Q: Y Show that f (2)= 2+ Con formal maps %3D on %3D

A: Given that,f(z)=z+1zz=1

Q: Prove the Cauchy-Schwarz Inequality, ∣u ∙ v∣ ≤ ∣ u∣∣  ∣ v ∣∣.

A:

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

A:

Q: Ex: O calkulake th gui for the disanse formula in a barred cordihute system defined by R = Ax, whene…

A: Here, one matrix function, A=1-223 is given Let the X=a1b1a2b2 So, X¯=AX=a1b1a2b2.1-223…

Q: Q2. (e) Verify that the Cauchy-Schwarz inequality holds. įu=(-3,1,0), v=(2,-1,3) u u=(0,2,2,1),…

A:

Q: Prove by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

A:

Q: U = kuzx for 0 0, u(0, t) = u(L, t) = 0, u(x, 0) = x(L – x)

A: Given that, ut=kuxx , 0<x<L, t>0u0, t=uL,t=0ux, 0=xL-x This is  the heat conduction problem…

Q: Verify the Cauchy-Schwarz Inequality for u = (1, −1, 3) and v = (2, 0, −1).

A: u = (1, −1, 3) and v = (2, 0, −1) To verify the Cauchy-Schwartz Inequality for the given set of…

Q: Let u = and v =| [:]} Use the Cauchy-Schwarz inequality to show that a² + b? 2 + (*) a +b 2 VI

A:

Q: of at (x, y, z) = (2, 3, 4). ay Let f(x, y, z) = (x + y)z. Evaluate

A: Given :fx,y,z = x+yz

Q: 4:x→X (when X is a normed Space) is continous at xo € X iff for any Sequeue ie AS n s

A: Let f:x→X (when X is a normed space) is continuous at x0 ∈X iff for any sequence  xn ∈X, xn→х0  ⇒…

Q: followinp serves Dedermine is he conditionelly , I converye your work ond justify your absdlutely…

A:

Q: (b) Show that x. y < ||x|| ||y||- [Hint: If x, y # 0, let a = , 6= and use the fact that ||ax + by |…

A: Given that, Euclidean metric d on Rn is defined as d(x,y) = || x - y || for x, y in Rn Part(a): Let…

Q: graph the inequality on a coodinate plane x &gt; with line under it 2

A: To graph the given inequality x≥2.The equation x=2 has a graph that is a vertical line with all…

Q: X Y Z' WXXN W. Y ED O XYZ' + XY + WX'YZ (XYZ)+(X+Y) + WX'YZ F

A:

Q: 2. Use the Squeeze Theorem to show that f(r) = 2 is in R(0, 3).

A:

Q: Proof by contrapositive that for x, y ∈ Z, if xy is even then either x or y is even.

A:

Q: A(n)............ is the collection of all points in the plane the sum of whose distances from two…

A: It is given that,   A(n) ............ is the collection of all points in the plane the sum of…

Q: b)use Cauchy Schwarz inequality to show that if Let u = and v = inequality to show that a² + b?

A:

Q: let (2, m) be a meesurable space. then 1f f9: x LJare measural ble function, then -> ) f43 is mea…

A:

Q: 28. Bounded by the planes z = x, y = x, x + y = 2, and z

A: Solution:-

Q: Let f(z, y, z) = z2 +y? + z²,z = /utv, y = /u – v, and z= uv, Find in terms of u and v.

A: we have to partial derivative with respect to u by using chain rule

Q: In Let x, y ER" be given. (a) If ||x|| = ||y|| = 1 and x - y = 1, show that x = y. (b) If ||x|| ≤ 1…

A:

Q: Find the bounded อน əx solution of อน 2 + U if U(x, 0) = 6e-3x. at

A: We have ∂u∂x=2∂u∂t+u  The initial condition is ux,0=6e-3x Now, ∂u∂x=2∂u∂t+u can be written as…

Q: Recall that RK denotes the real line with the K-topology. Let Y denote the quotient space obtained…

A: Given:-             Note that if U opens in R, and then U-K is opens in R k.             U = union…

Q: 2. Explain why there does not exist a measure Space (X,S, M) with the pro porty %3D

A:

Q: Q2. (e) Verify that the Cauchy-Schwarz inequality holds. įu=(-3,1,0), v=(2,-1,3) uu=(0,2,2,1),…

A: Cauchy-Schwarz inequality relates the inner product of two vectors and their norms.

Q: 2. Show using only the Cauchy-Schwarz inequality that if u,vE R", then || u- v ISl u || +| v | (1

A: Solution

Q: In general, for a a) S is compact 2) Sis closed at S is compact

A: We use definition of compact metric space, totally bounded set, Closed set, accumulation point or…

Q: 3 We can de fine the fallaing tunction on real mumbers ( called "absohite value") ; abs (x) = x if…

A:

Q: Theorem:For all Xiyiz,w EN i) X<y ^ z sw =)x+Z <ytw ii) xsy ^ zw =) x+Z< y tw )x<う tsw 1 Esw うメ+ と x…

A: Solution: We know that for x,y∈ℕ If x&lt;y then ∃k∈ℤ+,k&gt;0 such that x+k=y If x≤y then ∃k∈ℤ+,…

Q: Show that both ℝ and ℝ2 may be covered by countably many open balls

A:

Q: Find the unique fixed prbability vecter for the regular stochostic matrx A= ) 2 5 2 0 3 3

A: The following information has been given: The regular stochastic matrix is : A=311252033 We need to…

Q: 2 - Find the inter val on which f are Con Cave down and infleobion Points a -y = x2 5X+10 6- ソ= (¥+…

A:

Q: 10. Prove that the set H={[,ne subgroap of GL(2, R). For show that x HxcH but xHx+H. :9 GL(2,R),

A:

Power Operation

Power operation is topic of algebra in Math. It is use to represent repeated multiplication. Very big number and very small number can be easily express using power operation. Power operation is useful in many fields. In space engineering, it helps in representing the distance or size of particular heavenly body. In medical field, it is used to represent very small size. In medical field it helps to mention size of bacteria or virus. 

Exponents

The exponent or power or index of a variable/number is the number of times that variable/number is multiplied by itself.