(b) Given  = (x²y)î + (xz – z²)f – (xyz)k, determine the following at point (1, 1, 2): (i) V- Â. (ii) grad(div Â). (iii) curl(curl Â).

Q: Find the flux of F¯ = (x − y)i + xj across the circle x 2 + y 2 = 4 in the xy-plane

A:

Q: Integrate F(r)= [e2*, e²y,e²²] along the helix that cuts the z-axis at c.

A:

Q: For F(x, y, z) = x²z² i+ sinx z²j+zx²y*k, find div F and curl F.

A:

Q: Find a direction in the xy-plane in which f(x, y) = e¹-xy has zero change at the point P(1.0. e).

A: We know directional derivative formula Du→fa, b, c=∇fa, b, c.u^ Lets say u→=cosθ, sinθ be the…

Q: Sketch the slope field for y’ + y = 2 at the 25 gridpoints (x, y), where x = 0, 1, ... ,4 and y = 0,…

A: The given differential equation is y'+y=2. Sketch the slope field for the above equation as shown…

Q: (b)Find the derivative of g in the direction of a= 2i+ curl w where at point (0,-1,1) g = In- 4 + 2…

A: Given, w=cosxyzi+xeyz+1j+xyzk Then, Curl…

Q: Use the IVT to prove that the curves y = x2 and y = cos x intersect.

A:

Q: Find the maximum value of ƒ(x, y) = 6xye^(-(2x+3y) )in the closed first quadrant (includes the…

A: Given and

Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = 2xy3i…

A: Consider F = 2xy3i + 4x2y2j We have M = 2xy3 and N = 4x2y2

Q: of the tangent plane to the surface z=e-4x/17ln(1y) at the point (-3, 4, 2.808).

A:

Q: 2. Find the divections..in.w.hic . fCx,y) = X- a.. Increases.ms..rapidly.at. paint.C1). b. Decrenses…

A: Solution: The given function is fx,y=x22+y22    ......(1)

Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = (4x -…

A:

Q: Find the asymptoles of the funchon given Hx)= x²- 5x+6 below X-2

A:

Q: (5xy --- 6x(2^

A: Given,        F=(5xy-6x2)i+(2y-4x)j and   y=x3 from the point (1, 1) to (2, 8).

Q: Integrate counterclockwise dz by Cauchy's formula z2 – 16 counterclockwise around the circle C : |z…

A:

Q: what is the area of the rotational surface formed by rotating the part of the function y= x^3/3…

A: We apply the above formula to solve this

Q: Determine the curl and divergence of F(XYZ) = xye* ^-x³yze²^ + + xy²e¹ k

A: To find- Determine the curl and divergence of Fx, y, z = xyex i^ - x3yzez j^ + xy2ey k^.

Q: Find the points on the surface x² + y² + z² = 4 farthest from the point (1,–1,1), 2 %D

A:

Q: - Find div F and curl F if F(x, y, z) = 10y¹z7i – 17xªzºj – 3xyễk. div F= curl F=

A: The general form of a vector-valued function in three dimensions is given by F=F1i+F2j+F3k, where…

Q: 1. Let F(x, y) = 1 0 if x + 2y ≥ 1 if x + 2y < 1 Does F defines the DF in the plane?

A:

Q: Find the linearization L(x,y) of the function f(x,y) = e 3× cos (9y) at the points (0,0) and 0,

A:

Q: Find div F and curl F if F(x, y, z) = 13y'z4i – 16x4z10j – 3xy'k. - - div F= curl F=

A:

Q: Find the work done by F = 2xy°i + 9x<y*j in moving a particle once counterclockwise around the curve…

A: The solution is given below:

Q: 8. Calculate fo F • dr, where F(x, y) = y² i + (6x + 6y)j, and C is the portion of the parabola 2y²…

A: Given curve is 2y2-x=1 and points are (1,-1) to (7,2)

Q: Evaluate  ∫C (3y + exsinx) dx + (8x - ln(y3+2)) dy, where C is the circle x2 + y2 = 9  with positive…

A:

Q: 6. (a) Show that F = (3a²y+2, x3 + 3) is conservative. (b) Evaluate Sc(3x²y +2)dx + (x³ + 3)dy where…

A: According to the problem, we have

Q: (2) Find the Ualue of the constant m while the Curl of Foaoy,z) = (mxy- 2) Ĉ +cm-2)x)+(1-m)x2° Â is…

A: Please see the answer in step 2

Q: At what point on the paraboloid  y= x^2 +z^2is the tangent plane parallel to the plane x+2y+3z=1 ?

A:

Q: 4. Evaluate √3x²e¹² da + 2x¹yev dy where the open curve C is the semicircle x² + y² = 5, y ≥ 0 going…

A: 4. The given integral is ∫C 3x2ey2dx +2x3yey2dy where C is the semicircle x2+y2=5, y≥0 To find: the…

Q: (b) Use Leibniz's product rule to derive the quotient rule y du- v dy d YY

A:

Q: Find an equation for the tangent plane to the graph of z = xe' cos z – 2 - at the point (2,0,0). 1.…

A:

Q: The direchional derivative of the salan funchion at (l,-1,0) in the direction (i-j+2k) is xy- z- xyz…

A: Given that  Let f(x,y,z)=x^2y-y^2z-xyz To find the directional derivative at point

Q: 3y², Compute divà and curlà for à = (x²y²z?)î – (2xy² – 16x)ĵ + (9_)&.

A:

Q: Q2 A) for F= 2xi-3Y3+K %3D prove That div( curl È)=0

A: Given :          F→ = 2x i - 3y j + k   To prove that div(curl F→ ) = 0

Q: Find the Direcctional derivatve of the function foxxy,z)=tan(xjz) aut point (t,2,4) in the direction…

A:

Q: Find the linearization L(x,y) of the function f(x,y) = e* cos (9y) at the points (0,0) and ...

A:

Q: Find the points on the surface 1x^2+2y^2+5z^2=1 at which the tangent plane is parallel to the plane…

A:

Q: Evaluate F.dr, where F(x,y)=[2y-3cos(x)]i + [4x +6sin(y)]j and C goes counterclockwise around the…

A: The objective is to evaluate ∫CF·dr where Fx,y=2y-3cosxi+4x+6sinyj and C goes counterclockwise…

Q: Express x = In(3t), y = 5 – t in the form y = f(x) by eliminating the parameter. %3D y =

A: Given that x=ln(3t) and y= 5-t we need to write y=f(x) by eliminating the parameter  Now , x= ln(3t)…

Q: 3) Find the divergence and curl of f3xi+ 5x} j + 2y²k, at the point (1,2,3)

A: Solution is done in step 2

Q: Find the avea of the Vegion in the first quadrant between the Curles y =in X e y =2 inx g the line…

A:

Q: dx Calculate for n = 6, using trapezoidal rule. %3D -11+ x²

A:

Q: FIND V= (eX*z) čx + 2xy Èy+(xyi z³) &z THE CURL OF

A:

Q: Find the Võlumé of thé šólid generated by revolving the shaded region about the given axi About the…

A: Explanation: If we rotate the shaded region about the x-axis, we will get the cone shape. Height of…

Q: Find the maximum value of f(x, y) = 4xy(9 – 72x – 9y) over the triangle with vertices -- - (0,0), (5…

A: Given: fx,y=4xy9-72x-9y vertices 0,0 18,00,1

Q: Find the image of the line z =3 +iy under f(z) = e

A:

Q: Find the flux of F = (x - y)i + xj across the circle x2 + y2 = 1 in the xy-plane.

A: Given F = (x - y)i + xj  across circle x2+y2=1

Q: Ext find the anen of the ragion bounded on the leftby the Curve yx, on the right by the line y and…

A:

Q: draw the slope field for y' = x² – y², then use it to sketch three solution curves. %|

A:

Q: Find an equation for the tangent plane to z = 9x? – 8y² at the point (1, 1, 1). - = 0.

A: Here we find the equation of tangent plane.

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.