### Calculating the Line Integral of a Vector FieldLet $ C $ be the boundary of the quadrilateral with vertices $ (1, 1) $, $ (1, 2) $, $ (2, 3) $, and $ (2, 1) $ oriented clockwise. Let a vector field be given by:\[ F(x, y) = \begin{pmatrix}e^{x^4} + y^2 \\xy + \sin(\ln y)\end{pmatrix} \]**Objective:**Evaluate the line integral:\[ \int_C F \cdot dr \]**Instructions:**Show all supporting work.### Step-by-Step Solution:1. **Identify the Path $C$:** - The path $ C $ consists of four line segments joining the given vertices in the following order: - From $ (1, 1) $ to $ (1, 2) $ - From $ (1, 2) $ to $ (2, 3) $ - From $ (2, 3) $ to $ (2, 1) $ - From $ (2, 1) $ to $ (1, 1) $2. **Parameterize Each Line Segment:** - For each segment, create a parameterization using a parameter $ t $ ranging from 0 to 1.3. **Compute $ F \cdot dr $:** - Calculate the dot product of the vector field $F(x, y)$ with the tangent vector $dr$.4. **Integrate Over Each Segment:** - Perform the line integral on each segment and sum the results to find the total value of the line integral.### Supporting Work:- Ensure that each step is clearly shown, including the parameterization, calculation of $dr$, and evaluation of the integral for each line segment.- Sum the results of the integrals to obtain the final value.#### Detailed Discussion:- **Parameterization Example:** - For the line segment from $ (1, 1) $ to $ (1, 2) $: \[ x(t) = 1, \quad y(t) = 1 + t, \quad 0 \le t \le 1 \] - Here, \(

Answered: Image /qna-images/answer/1340c2ef-65a3-4dc3-9d49-a113df9f78d4.jpg

Let C be the boundary of the quadrilateral with vertices (1, 1), (1, 2), (2, 3) and (2, 1) oriented clockwise. Let a vector field be given by F (z, y) =< e²' + y², xy+ sin(ln y) > Evaluate: fa F. dr. Show all supporting work.

Let C be the boundary of the quadrilateral with vertices (1, 1), (1, 2), (2, 3) and (2, 1) oriented clockwise. Let a vector field be given by F (z, y) =< e²' + y², xy+ sin(ln y) > Evaluate: fa F. dr. Show all supporting work.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Related questions

Q: Find div F and curl F if F(x, y, z) = x²i – 5j + yzk. div F= curl F=

Q: Find the normal vector to the plane that contains the point (1, 1, 3) and the line r(t)=(0,1,…

A: Let P be the point 1,1,3.Q be the point 0,1,2 on the line.Then, PQ→ = 0-1,1-1,2-3=-1,0,-1And the…

Q: What is the flux through the described surface?

A: The surface is described by The vector field is F=<1, 3x, 2y>To evaluate the Flux for the…

Q: Q1/ If f(x,y)=find a unit vector u for which Duf (2,3) = 0 x+y

A: We have to find a unit vector u for which Duf(2,3)=0 where f(x,y)=yx+y

Q: Find a vector equation with parameter tt for the line of intersection of the planes x+y+z=4x+y+z=4…

A: What is Plane: A plane is a flat, two-dimensional surface that never ends in mathematics. A plane is…

Q: Find the directional derative of f at the point P0(1,1) in the direction of the vector A = 3i - 2j…

A: Direction derivatives at (1,1) is given in direction of A = 3i - 2j.

Q: H=pz Csup ap te'sin dg tpan 2 At point (1, ,6)

Q: Find the vector parametrization r(t) of the line L that passes through the points (2, 2, 5) and (6,…

Q: QI-A) Fnd the position, vekocity and the speed of the vector function ( r(t) ) that given bekow at…

A: Given vector function is r(t)→=t2i-t3j To find the position, velocity and the speed of the vector…

Q: -y+25 Let F (x, y, z) := x+yz7 3+2⁹ let n be the outward unit normal vector. Evaluate Sf (curl F) ·…

Q: Evaluate :// F. d5 , where F = and S is the helicoid with vector equation r(u, v) upward…

Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: Let f(x, y) = x^2/(x + y). Find a unit vector u such that Duf(1, 2) = 0

A: f(x, y) = x^2/(x + y)

Q: Let f(x,y, z) = a² + 2y² +z² and g(x,y,z)=z-xy. Find a unit vector u satisfying D.f (1.1,2)= 0 = D,…

Q: Q1] Find a unit vector normal to the surface 3x2 у — узz? аt the point (1,-2, -1)

Q: Find an expression for a unit vector normal, no, to the surface x = (4 – cos (v)) cos (u), y = (4 –…

A: The complete solutions are given below

Q: Find a vector normal to the surface 3z 3 + x 2 y - y 2 x = 1 at P =(1,-1,1) .

A: Given: 3z3+x2y-y2x=1 at P(1,-1,1)

Q: Find the flux of the vector field V(x, y, z) = 2xy²i+ 5x²yj + 3z³k out of the unit sphere. Flux =

A: Denote the unit sphere by S If the sphere is closed, then the flux of v(x, y, z) out of S is given…

Q: calculate div(F) and curl(F). F = (xy, yz, y² – x³)

Q: Find T, N, and B for the curve r(t) = (14t, 14t, e') at (0, 0, 1). %3D Hint: After finding T', plug…

Q: YouTrylt 4 Find a vector function r(t) that parameterizes the intersection of the elliptic cyl 9 4…

Q: Find a vector perpendicular to the surface described by the following equation at the point 2zy= =…

Q: (a) Let V = Rª and let 3 be the standard dot product on V. (i) Calculate B((1,0, 1,0), (0, 1,0, 1)).…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Show that the vector-valued function shown below describes the function of a particle moving in a…

A: Part - 1 : Given vector-valued function is rt = 5i +5j +3k +…

Q: For the scalar function f, div(grad f) is equal to Select one: a. + + dx?' ay? ' az? b. dx dy dz C.…

Q: Consider the function f(z,y, 2) ,>0, y>0 and the point P(1, 1, 2). a): Find the direction in which f…

Q: Write formulas for the unit normal and binormal vectors of a smooth space curve r(t)

A: The formulas for unit normal vector and binomial vectors of a smooth space curve r(t).

Q: 2. Let x'= [5, 1, 3] and y' =[-1, 3, 1]. (a) Find the inner product, y'x. (b) Find x- - 2y. (c) Find…

Question

### Calculating the Line Integral of a Vector Field

Let $ C $ be the boundary of the quadrilateral with vertices $ (1, 1) $, $ (1, 2) $, $ (2, 3) $, and $ (2, 1) $ oriented clockwise. Let a vector field be given by:

\[ F(x, y) = \begin{pmatrix}
e^{x^4} + y^2 \\
xy + \sin(\ln y)
\end{pmatrix} \]

**Objective:**

Evaluate the line integral:

\[ \int_C F \cdot dr \]

**Instructions:**

Show all supporting work.

### Step-by-Step Solution:

1. **Identify the Path $C$:**
- The path $ C $ consists of four line segments joining the given vertices in the following order:
- From $ (1, 1) $ to $ (1, 2) $
- From $ (1, 2) $ to $ (2, 3) $
- From $ (2, 3) $ to $ (2, 1) $
- From $ (2, 1) $ to $ (1, 1) $

2. **Parameterize Each Line Segment:**
- For each segment, create a parameterization using a parameter $ t $ ranging from 0 to 1.

3. **Compute $ F \cdot dr $:**
- Calculate the dot product of the vector field $F(x, y)$ with the tangent vector $dr$.

4. **Integrate Over Each Segment:**
- Perform the line integral on each segment and sum the results to find the total value of the line integral.

### Supporting Work:

- Ensure that each step is clearly shown, including the parameterization, calculation of $dr$, and evaluation of the integral for each line segment.
- Sum the results of the integrals to obtain the final value.

#### Detailed Discussion:

- **Parameterization Example:**
- For the line segment from $ (1, 1) $ to $ (1, 2) $:
\[ x(t) = 1, \quad y(t) = 1 + t, \quad 0 \le t \le 1 \]
- Here, \(

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Trending now

This is a popular solution!

Step by step

Solved in 4 steps with 4 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differentiation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions

7. On a two-dimensional plane, sketch the vector |v>= such that it emanates from the coordinate (4,5). [3]
None
Consider z as a function of the variables x, y which is denoted by z = In(x°y*) + x2 y5 and let P(1,-1) be a point. Consider further the vectorsu = (a, B), w = (-a, B), such that ||u|l = v 73. V73. W 79/73 97/73 If Duz(P) = 73 Duz(P) : and determine: %3D 73 The value of B:
sketch properly and provide full solutions! will give thumbs up!
191AfA, B are two differentiable vector point functions then prove that curl(A × B) = A (div B) – B (div A) + (B.V) A - (A V) B, %3D
Find a vector valued function that represents the intersection of a sphere w/ a radius of √68 and a plane with a trace of y=x^3 on the xy plane that extends orthogonally into the positive z. Sketch the curve created by the vector value function including the sphere and the plane. A computer generated sketch w/ annotation is acceptable.
9.Vector analysis (calculus 4)