Q. find the angle between the tangent vectors to the curves traced out by r¡(t1) and r2(t2) at the point of intersection given. ri (t1) r2 (12) = (1 – t2)i – j+ (t2 – 1)k at (0, –1, 0) (e'1 – 1)i – cos (Tt1)j+ tk and |

Q: Find the derivative of the function h (x, y, z) = cos xy+ e+ In zx %3D at P. (1, 0, 1/2), in the…

A: Here we have to find directional derivative of h(x,y,z) in the direction of u.

Q: 1. If the parametric curve $x=f(t), y=g(t)$ satisfies $g^{\prime}(1)=0,$ then it has a horizontal…

A: TRUE Slope of the curve, when in parametric form = dy/dx = g'(t) / f'(t)

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

A:

Q: (x cos-1(yz – z²) ту? xZ – 1 n( i+ mxy²j + y(z – mx)k \z² + m/ f = w • 1) if w = Find at point (-m,…

A: w¯=lnxz-1z2+m.i^+mxy2 j^+y(z-mx).k^andf=w¯.xcos-1(yz-z2)my2 j^f=xz-1z2+m.i^+mxy2…

Q: Find the function f if. Vf-(3a²y + y², a² + 2xy), Vf dr with and without the Fundamental Theorem of…

A:

Q: Q9. Find the equation of the tangent plane to f(x, y) = x ln(y) +x²y+3 at the point (x, y) = (2, 1).…

A:

Q: 1. Let f(x, y) = -x2 – xy³ + 7. Find an equation for the plane tangent to f at (3, 2).

A:

Q: 6: Find the work done by the force F = -7y i+ 10x j along one loop of the curver = sin 70 (in the…

A: Given

Q: find a parametrization c(t) of the curve satisfying the given condition.  y = x 2, c(0) = (3, 9)

A: The given function is y=x2 and c(0) =(3,9) Let x(0) =3 and y(0)= 9 Take x =t +x(0) Thus, x = t+3

Q: 3. Find the unit tangent vector at t = T/2 for the following parametrized curve: F(t) = (cos 2t, 4,…

A: The given parametrized curve r→(t)=cos2t, 4, sin2t

Q: The tangent line to the curve r(t) =(t°,t“, t³ ) at the point (1, 1, 1) intersects the æy-plane at a…

A:

Q: find the decomposition of a(t) into tangential and normal components at the point indicated, as in…

A: Consider the given information rt=eti+1-tj ,  t=0   Find the first derivative of rt…

Q: find the decomposition of a(t) into tangential and normal components at the point indicated r(t) =…

A: The given vector  rt=t,cost,sint ,      t=π2 To finding the components of acceleration take the…

Q: find a parametrization c(t) of the curve satisfying the given condition. y = 3x − 4, c(0) = (2, 2)

A: It is given that,

Q: 6. Find F' and dF[a; h] where: In ¤1 (a) F(x1,x2) and hence show that if v is tangent to the curve…

A: (a.) Given: Fx1,x2=lnx1x2e2x1x22cosx1 To find: F' and dFa;h To show: If v is tangent to the curve…

Q: Let z(t) = 5 cos (t) and y(t) = 3 sin(t) ELLIPSE 2- At t = 3, x(3) = -495 -1- y(3) = 0.42 -2- dz…

A:

Q: Suppose the vector-valued function r(t) = (f(t),g(t),h(t)) is smooth on an interval containing the…

A:

Q: find the decomposition of a(t) into tangential and normal components at the point indicated, as in…

A: Given: r(t)=(t,cost,sint)t=π2

Q: 2. Let F(t) = (e' cos t) î+ (e' sin t)j+ek. a. Find symmetric equations of the line tangent to the…

A: Given:  F→t = et costi^ +et sint j^ +et k^  To Find:  Symmetric equations of the line tangent to the…

Q: Compute I = (2x²132²y+ 4xy)ax +(x'+2x²-y') dy+zdz along The curve C C4U Cz UG given in the figure…

A: Given To compute I=∫C2x3+3x2y+4xy.dx+x3+2x2-y2dy+zdz along the curve C=C1∪C2∪C3 given in the figure.

Q: .: Evaluate [G(x, y)dx, [G(x,y)dy and [G(x, y)ds on the indicated curve C. G (x, y) = x³ y² + y² sin…

A:

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

A:

Q: find a parametrization c(t) of the curve satisfying the given condition.  y = 3x − 4, c(3) = (2, 2)

A: To find the parametrization

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 work done by F in moving a particle once counterclockwise around the given curve C F =…

A:

Q: Show that the function f(x, y) = e-2y cos 2.r satisfies the Laplace equation.

A: use partial derivative to check whether it  satisfy the equation or not

Q: Evaluate F. dr where F = (x – 3y)i + (y – 2r)j and C is the closed curve in the xy plane, x = 2 cos…

A:

Q: Find the value of the derivative dy/dx at  t=2π/3 when an object moves according to the parametric…

A:

Q: Consider the closed curve in the xy-plane given by x² + 2x + y* + 4y = 5. dy -(x+1) (a) Show that dx…

A: Given data:   The expression for the given function is x2+2x+y4+4y=5.   (a) Differentiate the above…

Q: Find the work done by F over the curve in the direction of increasing t. F= 11zi + 11xj + 11yk, r(t)…

A: We have to find the work done by F over the curve

Q: 2. Find an equation for the plane tangent to the surface 3x2 - 4xy+ z2 = 0 at the point (1, 1, 1).…

A:

Q: :. Let x(t) = 2 sinh t , y(t) = cosh(t²) be a curve in the Oxy-plane, given in parametric form. (a)…

A:

Q: 11. Compute the length of the curve r(t) = (2t, In(t), t²) over the interval 1 <t < 4. (cos(t),…

A:

Q: Find the equation of the tangent line to the space curve r(t) = 2cos (t)i + 2 sin(t)j + tk at the…

A: The curve is given by: r(t)=2cos(t)i+2sin(t)j+tk

Q: 1. If F(x, y) = (3+2xy)i + (x² – 3y²)j. Evaluate F. dr, where C is the curve given by r(t) = e'…

A:

Q: 5. Let f(1, y) = re" + cos(ry) and P: (2,0). A. Find the directional derivative of f at P in the…

A: Given information: The function is f(x,y)=xexy+cosxy and P:(2,0) and the vector is v=3i-4j. (a) To…

Q: Find a parametrization of the tangent line to r(t) = (ln(t))i + t^-5j + 5tk at the point t = 1

A:

Q: Find the work done by F over the curve in the direction of increasing t. F=xyi + 8j + 6xk; C: r(t) =…

A:

Q: b/ Show that the function satisfy Laplace equation f(x, y) In /9xy+3y2

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: 1. Find the tangent vector to the curve 6t r(t) = [t³ – t, t+1 , (2t + 1)*] at t = 1.

A: As per our guidelines we are supposed to answer only one question. If you want solution of any…

Q: 2. If F(x, y, z) = y²i+(2xy+e3=)j+3ye³=k. Evaluate F·Î ds, where C is the curve represented by r(t)…

A: Given F→x,y,z=y2i^+2xy+e3zj^+3ye3zk^ and the curve is C given by r^t=cost,t,sint,  0≤t≤π2. Find…

Q: b/ Show that the function satisfy Laplace equation f(x,y) = In /9xy+ 3y² %3D

A: Since you have asked multiple questions in a single request so we will be answering only first…

Q: Show that the curve r(t) = ( t3 /4 - 2)i +(4/t-3)j+cos(t-2)k is tangent to the surface x3 + y3 + z3…

A:

Q: Find the equation of the tangent plane to z = e² + y+ y*+3 at the point (0, 5, 634).

A:

Q: 1. Find the tangent vector to the curve 6t r(t) = [t° – t, (2t + 1)*] at t = 1.

A: