)=ス d²A A) dt dt dt2 dt dt3 dB dà xB) = Ã×: d?B d?A B) dt dt dt dt2 dt2

Q: - || e dA, where D is the open disk D = {(x, y) : x² + y² < 4}.

A:

Q: In AABC, X is the centroid. If CW = 15 , find XW. %3D

A: Given,           In △ABC, X is the centroid and CW¯=15

Q: In Exercises 11-28, use Separation of Variables to find the generalsolution. dy - = ytant dt

A: The given differential equation is dy=-ytantdt

Q: If KN=2x+1 and NJ=3x+5, what value of x makes point N the centroid? K, J M

A: (1).   If in the given triangle N is the centroid then the point N divides the line HL, JK and LM in…

Q: z dv, where E is enclosed by the paraboloid z = x2 + y² and the plane z = 4. Evaluate

A: see 2nd step

Q: Use Green's Theorem to evaluate the line integral. cx2 - y?) dx + 5y? dy C: x² + y² = 9 (x² .

A:

Q: Show that the function u e(xy) + Vry V(y/x) provides the quation = 0. dy?

A: Again differentiate,

Q: y dV, where B is the portion of the unit ball B = {(x, y, z) : x² +y² + z² < 1} which lies below the…

A:

Q: evaluate by Sto ke's theorom Gcyzdx+ zx dy+ xydz),where Cis the Corve xày=1, z=y: [ Hint: §…

A: Solution:-

Q: Let z = e* sin(y+a), show that %3D ôx? oy?

A: Given, z=e-xsiny+a, then we have to show that ∂2z∂x2+∂2z∂y2=0.

Q: Exercise 1 Use rectangular coordinates to evaluate [[[ y dV , where E is bounded by the planes x= 0,…

A:

Q: In AABC, X is a centroid. If AX = 6 units, calculate XE. E D B A

A: topic - centroid of a triangle

Q: Evaluate e dV where E is enclosed by the paraboloid z = 4+ x² + y?, the cylinder E x2 + y? = 1, and…

A:

Q: Find the volume of the parallelepiped (box) determined by u, v, and w while: u = 10i + 20j + 50k, v…

A:

Q: Determine la derivada direccional de f (x, y) = y In z en el punto P (1, 4) en la dirección de v = i…

A: Topic:- calculus

Q: The value of the line integral S(2xy dx + 2x?ydy + dz) C along a path joining the origin and the…

A: We have to find the value of given line integral.

Q: = a + zy", z - w* + w, y I = uu" + w", y u+ ve fi az Find when u = 1, v 2. w=0 = %3D H.

A:

Q: Show that the value of ∮C xy2 dx + (x2y + 2x) dy around any square depends only on the area of the…

A:

Q: du If u = x²y + y²z + z?x then at (1,1,0) is ax %3D -

A: Given that u=x2y+y2z+z2xTo find :∂u∂x  at 1,1,0Solution :u=x2y+y2z+z2x∂u∂x=2xy+0+z2∂u∂x=2xy+z2

Q: Evaluate the integrals

A:

Q: Evaluate ez dV where E is enclosed by the paraboloid z = 4 + x? + y², the cylinder x? + y? = 1, and…

A:

Q: Let B be the ball with center at the origin and radius 1 SS S6(x² + y² + z²)dV = 2 %3D B 12 167 24

A:

Q: Evaluate // e² dV where E is enclosed by the paraboloid z = 2 + x² + y², the cylinder E x² + y? = 3,…

A: Since we are given with the multiple number of questions and answered according to the guideline.

Q: soùd lines For Portions dash ùnes of borndry included in thee bmain and not include Suot1 For d)…

A: We are asked to find the domain of the function.

Q: The line integral 2xe" dæ + a²e" dy + (1 – 2²)e¬²² dz is not independent of path.

A: The line integral ∫CF·dr is independent of path if the curl of F is zero, that is, ∇×F=0. Here, The…

Q: 3. Find the circulation of F 2xi + 2zj +2yk around closed contour %3D

A:

Q: Transform the equation (ry? - z) dz + (z²y + y) dy = 0 into the form of separable dit %3D O ( + 1)…

A: Question is solved.

Q: Find the isogonal trojecdories of x*+y² =r² at 45",

A:

Q: Let v f(x,y) and w f(x, y). Choose a correct formula %3! to determine where J is a Jacobian. dv aw…

A:

Q: Find the outward flux across the domain D xi + yj + zk x2 +y? + z? F(x,v,z) = across the sphere x2…

A:

Q: U tt = quna %3D u(x, o) = Cos(x) Ut (X, o) = los (2x) %3D

A:

Q: Consider the following figure. R. Given: ST bisects RSV MERST - (x + y)e mzTSV = (2x - 3y)° !3!…

A: Given: The line ST bisects the ∠RSV. m∠RST=x+y°. m∠TSV=2x-3y°. m∠RSV=80°. Since, the line ST…

Q: Evaluate e dV where E is enclosed by the paraboloid z = 5 + x² + y°, the cylinder 2² + y? 1, and the…

A:

Q: se the Divergenci Theored tobevaluhbe Sis the bSH cordinate planes whe re bounded end the planess…

A:

Q: Compute where h(s) = rs? -r. Assume r is a constant. %3D ds

A: Since you have asked multiple questions in a single request, we would be answering only the first…

Q: Show that the function u = p(ry) + JFÿ¢(y/x) provides the quation „Pu = 0. dy

A:

Q: . Show that the function u = p(ry) + Vry v(y/x) provides the quation = 0. ду? %3D dx²

A:

Q: 2л where a, b > 0. dx = - ile* + b Prove that %3D

A:

Q: Evaluate fff(x +y +:' dV, where a is the solid hemisphere (half of the sphere) enclosed by x +y+ =1…

A: Here in the given question, find the integration of the given function:∭Dx2+y2+z232dVWhere Ω is the…

Q: 2. Evaluate the given line integral, (-y, x – y) · dR, where C is the line segment from (2,0) to (0,…

A: Using parametric equation, line integration is evaluated.

Q: Use Green's Theorem to calculate (422 – 3yx)dr + (3xy+ y² + 6)dy where C is the circumference (x –…

A:

Q: Integrate h (x. y) = yi+xj over the indicated path: r(u) = ui+ u²j. ue [0, 1).

A: Given hx,y=yi+xj and ru=ui+u2j, u∈0,1 We have to integrate hx,y=yi+xj along the path ru=ui+u2j…

Q: Evaluate e dV where E is enclosed by the paraboloid z = 2 + x² + y², the cylinder æ? + y² = 5, and…

A:

Q: Show that the function u = p(xy) + Vry v(y/x) provides the quation = 0. dy?

A:

Q: Use Green's theorem to evaluate s cy? oy)dx + (y²rlo4") dy where Cis s hoon belbw, N y = 2-x² G,1)

A: The given path contains 4 curves: C1:  from (1, 1)  to  (-1, 1)C2:  from (-1, 1)  to  (-1, -2)C3:…

Q: Evaluate e dV where E is enclosed by the paraboloid z = 4+x² + y°, the cylinder x² + y? = 3, and the…

A: Here we have to evaluate ∫∫∫E ez dV , where E is the enclosed by the paraboloid z=4+x2+y2 , the…

Q: Evaluate e dV where E is enclosed by the paraboloid z = 4 + x² + y², the %3D cylinder x? + y² = 3,…

A: We need to evaluate integral.

Q: Note: Answer either Yes or No or Not-Applicable in the block provided. Consider the DE ry" – y' +y =…

A: We will find out the required values.

Q: Qz: IF U=h cx,g )= Lnce+ ě).show that %3D Sy

A:

Q: Solve the fAllowing equetiun by cepuration of warnoble methal xQu-y Du t Zy = X Y glven uli,1)= 3…

A:

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.