Let F = (5y2, 3x2), and let C be the curve y = x-1 for 1 < x < 3, oriented from left to right. Calculate the dot product F(r(t)) · r'(t) dt for the parametrization of C given by r(t) = (t, t-1). F(r(t)) · r'(t) dt = dt

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A: The given function is fx,y=y2x and the ellipse 2x2+y2=1 Directed derivative,…

Q: B) Evaluate the line integral F-dr, where F (x, y) = r*y°i+x²y°j and C is the curve r (t) = (t³ +…

A:

Q: Evaluate the line integral I of the one-form w = xy dx + 12y dy over the parametric curve a : [0,…

A: Given: One-form ω=xydx+12ydy and the parametric curve α:0, ln2→ℝ2 defined by α(t)=e2t, e-2t. To…

Q: Exercise (2): Calculate S. xydx + 3y*dy where C is the curve in xy plane, x = y² from (0,0). to…

A: We have to find the line integral

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A:

Q: 3. Evaluate the line integral f. F· dr where F = (ev+ye")i+(xe"+e")j and C is the parameterized…

A:

Q: Find the parametrization of the tangent line to the space curve r(t) = (e" , sin(t), (t + 1)-³) at t…

A: Given that  r(t)=(e^(4t),sin(t),(t+1)^(-3)) at t=0 We have to find out the parameterization of the…

Q: Use Green's Theorem to find the work done by F(x,y)=x(3x+y)i+xy^2j in moving a particle from the…

A:

Q: Evaluate the line integral / V(ry + xz + yz) · dĩ over the curve F(t) = (t, 2t, 3t) for 0 <t < 4…

A:

Q: Find the derivative of ƒ(x, y, z) = xyz in the direction of the velocity vector of the helix r(t) =…

A: Concept: A vector is a directed line segment whose length is the magnitude of the vector and with an…

Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (b) Reparametrize R using…

A:

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A:

Q: Let f(x, y, z) = (2x + 4z + 10)/xy + 2. %3D a. Evaluate the line integral of f along the line…

A: We will find out the required solution.

Q: Use a scalar line integral to find the length of the curve r(t) = (20 sin , 20 cos ,5), for 0 < t <…

A: Given

Q: Find the work done under the influence of the force E = 3xyi – 5zj + 10xk along the curve x= t² + 1,…

A:

Q: Suppose F(x, y) = -yi + xj and C is the line segment from point P = (5,0) to Q = (0, 3). (a) Find a…

A: Given: F→(x, y)=-yi→+xj→For part(d)we know∫TF→.dr→=∫C1F→.dr→+∫C2F→.dr→+∫C3F→.dr→where T is the…

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A:

Q: Evaluate the line integral f. F. dr where F = (2 sin a,–5 cos y, 10xz) and C is the path given by…

A: We will evaluate the given line integral.

Q: Let f(x, y, z) = 3ex cos(yz). a) Find the directional derivative of f at P(0, 0, 0) in the…

A: Given , fx,y,z=3ex cos4zwe have to find directional derivative of f at p 0,0,0 in the directionof…

Q: Let f(x, y, z) = (2x + 4z + 10)/xy + 2. %3D a. Evaluate the line integral of f along the line…

A: We will evaluate the given integral.

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A:

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A: Given surface is z=xy

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A: coordinates of the centroid of the curve  x=cost+tsinty=sint-tcost…

Q: Suppose C is the curve r(t) = (6t,t°), for 0sts2, and F = (3x,3y). Evaluate F•Tds using the…

A:

Q: Evaluate the line integral F. dr, vhere F(z, y) = -yi-rj and C is the curve y = v4-a from (2,0) to…

A: We have to find the line integral

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A:

Q: Find the derivative of f (x, y,z) = xyz in the direction of the velocity vector of the helix…

A: Answer and explanation is given below...

Q: Compute the directional derivative of f(r. y) = = at the point (1, 1) in a irection normmal to the…

A:

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: Evaluate the line integral (2 - u) dz + (3 + æ) dy] where C is the parabolic arc = a² from (0, 0) to…

A: We have to evaluate the line integral.

Q: (a) Find a function f(x, y, z) which satisfies Vf = (y + 2z, æ + 3z, 2x + 3y) (b) If C' is the curve…

A:

Q: Evaluate the line integral F. dr, where F (x, y) = (3+2xy) i+ (x² – 3y²) j and C is the curve given…

A: Given problem:-

Q: Find the tangent plane for the level surface f(x,y,z) = 7x³ + cos (4y) + In (4z) = c at the point…

A: Tangent plane for the level surfacefx,y,z at (a, b, c) is…

Q: B) Evaluate the line integral F-dr, where F (x, y) = x²y³i+ x°y°j and C is the curve r (t) = (t³ –…

A:

Q: Calculate the integral ∫C(x2+y2)zds, where the curve C is parameterized by r(t)=(sin3t,cos3t,4t),…

A:

Q: Compute the line integral of the scalar function f(x,y,z)=2x2 +8z over the curve…

A:

Q: Suppose F(x, y) = x²i + yj and C is the line segment segment from point P = (2, –1) to Q = (4, –4).…

A: We need to evaluate integral.

Q: Calculate the work done by F(x,y) = ⟨2x + ln( 2y2 −1), arctan(y4 +2y +1) + 4xy / (2y2 − 1⟩ along the…

A:

Q: Suppose F(x, y) = -yi + xj and C is the line segment from point P (3,0) to Q = (0, 2). (a) Find a…

A:

Q: Use Green's theorem to evaluate the line Integral for the curve C given in the figure. [2y dx + x…

A:

Q: Use Green's theorem to evaluate (2y dx - x dy) around the closed curve C. y (2,5) 4 3 2 C 1. -2 -1…

A:

Q: Compute the line integral | (yze*v= + y° + z - 2x) dr + (xze*# + 2ry) dy + (rye#"= + x + 32²) dz,…

A:

Q: Use the Fundamental Theorem of Line Integrals to find the exact value of the path-independent line…

A: First of all hare we check for the path independent line integral then solve for moving over the…

Q: Consider the surface x°z5 + sin(y4z5) – 8 = 0. - Use implicit differentiation to find the following…

A: SOLUTION-

Q: Compute the line integral [(yze²v² + y² + 2 − 2x) dx + (xze²v² + 2xy) dy + (xye²³² +x+32²) dz, where…

A: We have to compute the line integral ∫Cyzexyz+y2+z−2xdx+xzexyz+2xydy+xyexyz+x+3z2dz, where C is the…

Q: if f(x,y) =x2y÷x+y and y(t)=  (x(t),y(t)) is parametrized curve in domain of f with y(0)=(2,-1) and…

A: The first step in solving the problem is to obtain the parametric equation. The initial values can…

Q: Suppose n is a vector normal to the tangent plane of the surface F(x, y, z) = 0 at a point. How is n…

A: We have asked how a vector n which is normal to the tangent plane of the surface F(x,y,z)=0 at a…

Q: Given: vector-valued function R(t) whose derivative R' (t) = (4t, 6t, -12t) (b) Reparametrize R…

A: Let's reparameterize vector value function R(t).

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.