Question

(12 points) Let $F(x, y) = 2xy^3 \mathbf{i} + (1 + 3x^2y^2)\mathbf{j}$. (a) Demonstrate that F is conservative. (b) Find the potential function $f$ such that $\nabla f = F$. (c) Evaluate the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ where C is given by $\mathbf{r}(t) = (t+1)\mathbf{i} + (2 - t^2)\mathbf{j}$ where $0 \le t \le 2$.

          (12 points) Let $F(x, y) = 2xy^3 \mathbf{i} + (1 + 3x^2y^2)\mathbf{j}$.
(a) Demonstrate that F is conservative.
(b) Find the potential function $f$ such that $\nabla f = F$.
(c) Evaluate the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ where C is given by $\mathbf{r}(t) = (t+1)\mathbf{i} + (2 - t^2)\mathbf{j}$ where $0 \le t \le 2$.

(12 points) Let F(x, y) = 2xy^3 𝐢 + (1 + 3x^2y^2)𝐣.
(a) Demonstrate that F is conservative.
(b) Find the potential function f such that ∇ f = F.
(c) Evaluate the line integral 𝐅· d𝐫 where C is given by 𝐫(t) = (t+1)𝐢 + (2 - t^2)𝐣 where 0 ≤ t ≤ 2.

Added by Joseph F.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

07/28/2023

Step 1

Step 1: To demonstrate that F is conservative, we need to show that the curl of F is zero. Show more…

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Text: (12 points) Let F = 2xyi + (1+3j). (a) Demonstrate that F is conservative. (b) Find the potential function f such that ∇f = F. (c) Evaluate the line integral ∫F · dr where C is given by r(t) = (t + 1)i + (2t^2)j where 0 ≤ t ≤ 2.