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 $\int_{\gamma} \mathbf F \cdot d\,\mathbf r$
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.
- answered
- 752 views
- $3.20
Related Questions
- Calculate $\iint_R (x+y)^2 e^{x-y}dx dy$ on the given region
- You have 100 feet of cardboard. You need to make a box with a square bottom, 4 sides, but no top.
- Fields and Galois theory
- Differentiate $f(x)=\int_{\sin x}^{2} \ln (\cos u) du$
- Evaluate the line intergral $\int_C (2x^3-y^3)dx+(x^3+y^3)dy$, and verify the Green's theorem
- Prove that $\int_0^1 \left| \frac{f''(x)}{f(x)} \right| dx \geq 4$, under the given conditions on $f(x)$
- Tensor Product II
- Urgency Can you help me Check these Applications of deritive.