1) Given F(x, y, z) = x³ i + 3yj - z4 k and the curve C parametrized by r(t) = (sint, cost,t), te [0, π] evaluate F.idx and F. k dz a) b) F.T ds c) Consider the closed curve C' obtained by the union of C and the segment that joins the points (0,-1, π) and (0,1,0). Evaluate the integral of F along C' and verify that it is equal to zero.
1) Given F(x, y, z) = x³ i + 3yj - z4 k and the curve C parametrized by r(t) = (sint, cost,t), te [0, π] evaluate F.idx and F. k dz a) b) F.T ds c) Consider the closed curve C' obtained by the union of C and the segment that joins the points (0,-1, π) and (0,1,0). Evaluate the integral of F along C' and verify that it is equal to zero.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Could you help me with these
![### Worksheet 18: Group Work
#### Problem 1:
Given \( \mathbf{F}(x, y, z) = x^3 \mathbf{i} + 3y \mathbf{j} - z^4 \mathbf{k} \) and the curve \( C \) parameterized by \( \mathbf{r}(t) = (\sin t , \cos t , t) \), \( t \in [0, \pi] \), evaluate the following:
a) \( \int_C \mathbf{F} \cdot \mathbf{i} \, dx \) and \( \int_C \mathbf{F} \cdot \mathbf{k} \, dz \)
b) \( \int_C \mathbf{F} \cdot T \, ds \)
c) Consider the closed curve \( C' \) obtained by the union of \( C \) and the segment that joins the points \((0,-1, \pi)\) and \((0,1,0)\). Evaluate the integral of \( \mathbf{F} \) along \( C' \) and verify that it is equal to zero.
#### Problem 2:
a) Show that any vector field in the form of \( \mathbf{F}(x, y, z) = f(x) \mathbf{i} + g(y) \mathbf{j} + h(z) \mathbf{k} \), where \( f \), \( g \), and \( h \) are continuous functions that depend only on one variable, is a gradient field.
b) Evaluate the integral of the vector field \( \mathbf{F}(x, y, z) = \sin x \, \mathbf{i} + \ln(y^2 + 1) \, \mathbf{j} \) over the “crazy curve” parameterized by
\[
\mathbf{r}(t) = \left( t(t - 1) e^{t^3 + 4 \sin t} / (1 + \cos^2 (4t - 7)) \right) \mathbf{i} + \left( t(t^2 - 1) (\tan t^3 + 4 \sin t) / (1 + e^{4t - 7}) \right) \mathbf{j}, \, t \in [0, 1]
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F304238c1-892d-4f66-90f4-8d5405bfbe58%2F77ab1302-2a49-4df1-9f18-d3a162b828b1%2F6la5r6b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Worksheet 18: Group Work
#### Problem 1:
Given \( \mathbf{F}(x, y, z) = x^3 \mathbf{i} + 3y \mathbf{j} - z^4 \mathbf{k} \) and the curve \( C \) parameterized by \( \mathbf{r}(t) = (\sin t , \cos t , t) \), \( t \in [0, \pi] \), evaluate the following:
a) \( \int_C \mathbf{F} \cdot \mathbf{i} \, dx \) and \( \int_C \mathbf{F} \cdot \mathbf{k} \, dz \)
b) \( \int_C \mathbf{F} \cdot T \, ds \)
c) Consider the closed curve \( C' \) obtained by the union of \( C \) and the segment that joins the points \((0,-1, \pi)\) and \((0,1,0)\). Evaluate the integral of \( \mathbf{F} \) along \( C' \) and verify that it is equal to zero.
#### Problem 2:
a) Show that any vector field in the form of \( \mathbf{F}(x, y, z) = f(x) \mathbf{i} + g(y) \mathbf{j} + h(z) \mathbf{k} \), where \( f \), \( g \), and \( h \) are continuous functions that depend only on one variable, is a gradient field.
b) Evaluate the integral of the vector field \( \mathbf{F}(x, y, z) = \sin x \, \mathbf{i} + \ln(y^2 + 1) \, \mathbf{j} \) over the “crazy curve” parameterized by
\[
\mathbf{r}(t) = \left( t(t - 1) e^{t^3 + 4 \sin t} / (1 + \cos^2 (4t - 7)) \right) \mathbf{i} + \left( t(t^2 - 1) (\tan t^3 + 4 \sin t) / (1 + e^{4t - 7}) \right) \mathbf{j}, \, t \in [0, 1]
\]
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!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
