Find a tangent vector of the curve7(t) = (t², 2 sin(t), 2 cos(t)) at (0,0, 2) (0,0,1) (1,0,0) None of the above or below (0,1,0) O (1,1,1) ) (1/2,0, 1/2)

Q: Use Green's Theorem to find f. [(2x – y+ 4) dx + (5y+ 3x – 6) dy] around a triangle in the xy plane…

A:

Q: Use Theorem 13.9 to find the directional derivative of the function at P in the direction of the…

A: Given function is, f(x,y)=sin(2x+y) and u=cos(-5π6)i+sin(-5π6)j⇒u=cos(5π6)i-sin(5π6)j=-32i-12j Now,…

Q: Find the unit tangent vector to the curve defined by F(t) = (4 cos(t), 4 sin(t), 3 sin²(t)) at t = 3…

A:

Q: Find the point on the graph of z = 3x2 + 2y at which vector (24, 8, 2) is normal to the tangent…

A:

Q: Verify Stokes's theorem for F =, where C is the curve C: x² + y² = 4, z = 2

A:

Q: Find the directional derivative of the function in the direction of the unit vector u = cos(0)i +…

A: Given- To do -

Q: Find the directional derivative of the function at the given point in the direction of the vector v.…

A:

Q: Find the curl of the vector field F (5x sin(y), 2y cos(æ)) curl F =

A:

Q: Find the vector equation of the tangent line to the curve given by 7(t) = sinti + cos (2t)j + tan tk…

A:

Q: Find the directional derivative of the function f(, y) = In(x + y°) at the point (- 1,3) in th…

A:

Q: Find the parametric equations of the tangent line to the curve represented by the vector function…

A:

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: /Find the tangent and normal vector for curve r= (5sin n i+5 cos n j+n k) n=(In 2t)

A:

Q: 1. Find the unit tangent vector T(t) and principal normal vector N(t) for the curve r(t) = (2…

A:

Q: 1. Find one normal vector of the tangent plane to f(x, y) = cos(x² + y²) at point P(1, 1, cos 2).

A: Please check step 2 for the solution.

Q: 1. Find one normal vector of the tangent plane to f(x, y) = cos(x² + y²) at point P(1, 1, cos 2). 2.…

A: 2. The equation of the tangent plane to f(x,y)=sin(x2+y) when x=0, y=π

Q: Find the equations of the normal line and the tangent plane to the parametric surface defined by the…

A:

Q: Find a vector equation of the tangent line of f(t) = (cost, sint, t) at the point Po (0, 1,

A:

Q: A normal vector to the tangent plane for the at the point (1, 1, 1) is: y surface x =

A: A(1, 1, 1)z= xyf(x,y,z) =xy-zfx=yfy=xfz=-1Normal vector to the tangent plane for the surface;n^…

Q: Find the unit tangent vector of the curve: 7(t) = (3 cos t , 3 sint, t2) at t = 0. %3D

A:

Q: Find one normal vector of the tangent plane to f(x, y) = cos(x² +y²) at point P(1,1, cos 2).

A: As we know the expression of the normal vector to the tangent plane at a point on the curve can be…

Q: Find the unit tangent vector to the curve defined by F(t) = (4 cos(t), 4 sin(t), 3 sin (t)) at t = 3…

A:

Q: FIND THE DIVERGENCE OF THE VECTOR F GIVEN = sin (8x)i + cos (5y); + z ² k FIELD F(X, Y, 2) BY

A:

Q: Sketch the vector-valued function r(t) = ⟨cos(t), 1, sin(t)⟩,

A: Sketch the vector-valued function r(t) indicating the orientation of the curve with an arrow.…

Q: 7. Find the directional derivative of the function f(x, y, z)=(y+28) at (2, 1, 1) in the direction…

A:

Q: Find the unit vector towards the tangent of the curve y = x at x =1

A:

Q: Find the tangent plane for the level surface f(x,y,z) = 3x5 + cos (5y) + In (5z) = c at the point…

A:

Q: Find the directional derivative of the scalar function of (x, y, z) = xyz in the direction of the…

A: Given surface is z=xy

Q: Find the normal vector to the tangent plane of z 7ea-2y at the point (4,8,7) x component = y…

A: The tangent plane is given as z=7ex2-2y and the point is (4,8,7) Let f(x,y,z)=7ex2-2y-z Then obtain…

Q: Sketch the space curve. Vector-Valued Function r(t) = (5 sin t, 2t, 5 cos t) Interval [0, π]

A: Given rt=5sint,2t,5cost So, r't=5cost,2,-5sint So,…

Q: Compute the principal unit tangent vector at t = T/2. F(t) = sin(2t) î + 2 cos(t) ĵ + tk COS 2 2 - -…

A: unit tangent vector

Q: 3. Find the directional derivative of f(x, y, z) =x*y-yz' +z at the point (1, -2,0) in the direction…

A:

Q: Find the directional derivative of the function f(x, Y, z) = x + ln(y² + z²) at the point A(2,1, 1)…

A: To find the directional derivative of the function f(x,y,z)=x+ln(y2+z2) at point A(2,1,1)  in the…

Q: Find the curl of the vector field F = (6y cos(x), 7x sin(y)) curl F =

A: If  F=f1x,y, f2x,y   thenCurl F=0,0,∂f2∂x-∂f1∂y

Q: Find the directional derivative of the scalar function of (x, y, z) = xyz in the direction of the…

A: What is Gradient of a Scalar Function: The gradient of a scalar function is vector quantity. It is…

Q: Find the curl of the vector field F = (3y cos(x), 5x sin(y))

A:

Q: Find the curl of the vector field F = (3x sin(y), 7y cos(x)).

A: The solution is:

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: Find the tangent plane for the level surface f(x,y,z) = 7x5 + cos (5y) + In (9z) = c at the point…

A: Answer: In oerder to use gradients we introduce a new variable c= 7x5+cos(5y)+ln(9z) Therefore the…

Q: 16. Sketch the cycloid r(t) = (t- sint, 1 - cos t) together with its tangent vectors at t = and .…

A: Solved the above problem.

Q: Find the flux out of the curve bounded by the top half of = arccos(√√²+² ) and y=0 with vector field…

A: As per the question we are given a bounded curve C and a 2d vector field F And we have to find the…

Q: (b)Find the directional derivative of f(x,y,z) = x²y- yz3 + z %3D at point (1,-2,0) in the direction…

A:

Q: Find a vector equation for the tangent line to the curve 7(t) = (5 cos(6t)) i + (5 sin(6t)) j +…

A:

Q: Example 4. Find the tangent vector at the point (2,4). on the top half of the circle x² + y² = 1…

A: Introduction: For a given vector curve rt, the equation that represents the tangent line to the…

Q: Choose the normal vector of the tangent plane to the surface defined by at the point (-4, 2, e88). O…

A: Given problem:- Find the normal vector of the tangent plane to the surface defined by…

Q: (b) Find the directional derivative of the function f(x,y,z) = xy sin z at the point (1,2, %3D T2)…

A: Solution:

Q: Find the derivative of alpha of t if alpha of t is the measure of the angle between the two given…

A:

Q: Compute the divergence and curl of the vector field cos (22 ) sin² (z² yz) 10 yz4 – xyz2 at the…

A: We have to find divergence and curl of vector field:

Q: Find the divergence and curl of the vector A = rz sin ø ↑ + 3rz² cos o and evaluate both answers at…

A: Consider the given vector. A=rzsinϕr^+3rz2cosϕϕ^ The divergence formula is defined as. divF=Px+Qy…

Q: Find the directional derivative of the function at the given point in the direction of vector v.…

A:

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: