Verify that wy =Wyx for w = In (4x + 7y).

Q: Compute Div(F) and Curl(F): F--(4y-1) i+yj+(-3y) k

A:

Q: Evaluate the line integral. | (9x – 9y) ds C C : x² + y² = 64 | (9x – 9y) ds = C

A: Explanation of the solution is given below...

Q: Find fx, fy and f z if f(x, y, z) = e*yz In zx. %3D

A:

Q: こ= (t)} () て= () %3D - (^),J %3D てんn ZトXC

A: →∫eax+bdx = eax+ba + C, where 'C' is the constant of integration.→∫xndx = xn+1n+1 + C

Q: Determine the Linearity of this DE. * ay dy ニ X Linear O Non-Linear

A:

Q: Find curl(Sz i+6x j + 7y k), curl(Sz i+ 6x j + 7y k) =( i+(

A: This question van be solved using the concept of curl F.

Q: If u =f(x² +2yz, y² + 2zx ).Prove that (v² – zx) + (x² - yz) ди + ди ху

A:

Q: ) If ƒ(x,y) = 3x²y + cosy+ ysinx – e2x then verify fxy = fyx

A: fx and fy are the partial deferential of function w.r.t x and y respectively. Then fxy is the…

Q: Prove that A <m AUB where AUB = {0x | r E A}U{ly|y € B}.

A: Solution: Let us consider two sets A and B. It may be possible that A∩B≠∅. But we want to construct…

Q: y z = x + show that : 0.

A:

Q: show that the scalar t that minimizes ||y– tx||

A: In this question, we need to find the t that minimizes ||y-tx||.

Q: 2x+ 3y-z =1 4x + y-3z = 11 3x-2y +5z = 21 | %3D

A: Solution:

Q: Compute the Wronskian of the following equation. y" – 4y' + 4y = xe2x %3D

A: Given: y''-4y'+4y = xe2x

Q: Obtain characteristics y² Uxx + Uyy = O

A: Given equation is  y2Uxx+Uyy=0 Now comparing with  AUxx+BUxy+CUyy=0 We have A=y2, B=0, C=1

Q: If FGx) = -2x+s firid f(z)

A: Given: if f(x)=-2x+54To find: f(1)

Q: AISO is isosceles with IS = IO. 22 = 2x+24, mZ3 = 6x– 52, %3D %3D nd mZ1. 1/ 3 bX 2 bx -52-180 S x-…

A: We have to solve the problem

Q: (a) 3y' In (u) dy

A: We will use the ILATE rule in this example to solve the integral and for solving we use the formula…

Q: For x,y E [0,1], prove that xy(1-x-y)≤1/27

A:

Q: fx) =x+x, aニー Lcx) = ? (メフフ %3D

A: Given that fx=x+1x,      a=−1

Q: Verify that ƒxy = ƒyx, for ƒ(x, y) = xey.

A: Given function is fx,y=xey We have to verify fxy=fyx Solution: fxy means that first we find partial…

Q: f(z) =z2 is discontinuous in C

A:

Q: Verify that wxy = wyx for w= In (4x + 5y). W. ху Wyx Compare wxy and wyx. Choose the correct answer…

A: Given query is to find the correct option for partial derivatives.

Q: () Let z = x + iy. Find Im and Im 2 in terms of x and y. Im Im z2 ||

A:

Q: Let f(x,y,z) = x2yz4 + 4xy²z+y²z - 3y³z Find fxyz

A:

Q: 2x+ 3y-z =1 4x + y-3z = 11 3x- 2y +5z =21

A:

Q: Compute ƒxz and ƒzz for ƒ(x, y, z) = xyz - x2 z + yz2.

A: Given that f(x,y,z)=xyz-x2z+yz2. We need to find : i)fxz. ii)fzz.

Q: find an antiderivative Fon of the function fox)=5sinX such that FCO)=1.

A:

Q: Express In x V in terms of in x, In y, and In z. 2mx-글 my-글 nz 2 Inx + 글 Iny-글 mz znx+my+nz In y 2nx…

A:

Q: Let F I2). Evaluate JJaw as z² + y? 0

A: In the given question, the concept of the Divergence Theorem is applied. Divergence Theorem Assume S…

Q: dt Determine dw if t = (6r* +3r*) -(*+2r-5)'. -s)'. and w =

A:

Q: Let z = 3x*y, find Vz.

A:

Q: Calculate A and B so that t B %3D 1+1 (1-z²)

A:

Q: bot A de groutne conitant. est Cheafuntis uh that f cark3 and f'4)=-s. g(xl=In (f(x)) + A conx)

A:

Q: Q2:show that the force Fexyzi-x4z'k is Conservative. ?

A:

Q: fx) =x+x, aニー %3D LCx) = ? %3D

A: Our objective is to find the linearization L(x) at x=a.

Q: ди Ifu=r(x* +2yz,y* +2zx)Prove that (y -zx) +(x² - 3z) + (:* - xy) ZX

A:

Q: xy Let z . Then: 4x2 – 5y? | az dz ду

A:

Q: Find (∂z)/(∂s) and (∂z)/(∂t), given: (a) z = x2 - 2y2 ; x = 3s + 2t, y = 3s - 2t

A: It is given that,            z = x2-2y2  ;  x = 3s+2t  ;   y = 3s-2t We have to find  ∂z∂s  and…

Q: Find f,, fy, and f, for f(x,y,z)= 5xy+ 3xz + yz.

A: It is partial derivative

Q: Find those constants a, b, c, d such that f(z) = ax² + bxy + y² + i(x² + cxy + dy?)

A: The given function is : fz=ax2+bxy+y2+i(x2+cxy+dy2)

Q: Let u = 2i−j+3k, v = 3i+j−k, and w = 3i+j+2k. Verify Equation (3).

A: Given- u=2i-j+3k,v=3i+j-k and w=3i+j+2k To verify- The equation u×v·w=u·v×w

Q: b) u(х, у) on the domain 0 < [z| < ∞. x² + y? Why is v(x, y) single-valued in b?

A:

Q: Let D, be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function.…

A: Dx(f)=8x+5 Apply integration both side  ∫Dx(f)dX=∫(8x+5)dx   f(x)=(8x2)/2 + 5x +c (here c is the…

Q: Is F(x,y)= (2²y,5– zy²) source free?

A:

Q: Verify that wy = w for w = In (6x + 5y). Wxy Wyx Compare wxy and wyx. Choose the correct answer…

A:

Q: Calculate az/ax, where xez + zeY = x + y.

A:

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: (b) Compute f er+uds, where c(t) = (t, 2t), t e [1, 2]. %3D

A:

Q: If z = u2 v3 w4 , u = t2 , v = 5t − 8 and w = t3 + t, find dz/dt

A: Given,           z = u2v3w4 , u = t2 , v = 5t − 8 and w = t3 + t

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.