Vector field
the vector field $F(x,y)= [ -2 \frac{x}{(x^{2} +y^{2}+1)^2 },-2\frac{y}{(x^{2}+y^{2}+1)^2 } ] $ is conservative throughout the planet. Determine a potential U to F and calculate $\int_{\gamma} \mathbf F \cdot d\,\mathbf r$ where γ is an arbitrary curve in the plane from (3,−4) to (4,1).
calculate U and $\int_{\gamma} \mathbf F \cdot d\,\mathbf r$
Asara
Answer
A potential is:
$$U(x,y) = \frac1{x^2+y^2+1}$$
as you may see via the chain rule:
$$\partial_x U(x,y) = -2\frac x{(x^2+y^2+1)^2},\qquad \partial_y U(x,y) = -2\frac y{(x^2+y^2+1)^2}$$
So for any curve $\gamma$ from $(3,-4)$ to $(1,4)$ we have:
$$\int_{\gamma} F\cdot d\gamma = \int_\gamma \mathrm{grad}(U)\cdot d\gamma = U(4,1)-U(3,-4) = \frac1{18}-\frac1{26}$$
by the fundamental theorem of calculus / Stokes' theorem, whatever you are using.
Dynkin
- answered
- 1337 views
- $3.20
Related Questions
- Algorithm for printing @ symbols
- Urgency Can you help me Check these Applications of deritive.
- Use Rouche’s Theorem to show that all roots of $z ^6 + (1 + i)z + 1 = 0$ lines inside the annulus $ \frac{1}{2} \leq |z| \leq \frac{5}{4}$
- Beginner Differential Equations - Growth Rate Question
- Population Equations
- Rose curve
- Prove that $\int_{-\infty}^{\infty}\frac{\cos ax}{x^4+1}dx=\frac{\pi}{2}e^{-\frac{a}{\sqrt{2}}}(\cos \frac{a}{\sqrt{2}}+\sin \frac{a}{\sqrt{2}} )$
- Velocity of a rock
