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Digiron [165]
1 year ago
10

Calculus 283 Chapter 16​

Mathematics
2 answers:
ycow [4]1 year ago
8 0

(a) The vector field \vec F is conservative if it is a gradient field. So we try to find a scalar function f such that

\nabla f(x,y) = \dfrac{\partial f}{\partial x} \, \vec\imath + \dfrac{\partial f}{\partial y} = \vec F(x,y)

Given the field

\vec F(x,y) = y\,\vec\imath+x\,\vec\jmath

we need to have

\dfrac{\partial f}{\partial x} = y

\dfrac{\partial f}{\partial y} = x

Integrate both sides of the first equation with respect to x.

\displaystyle \int \frac{\partial f}{\partial x} \, dx = \int y \, dx \implies f(x,y) = xy + g(y)

Differentiate both sides with respect to y.

\dfrac{\partial f}{\partial y} = x + \dfrac{dg}{dy} = x \implies \dfrac{dg}{dy} = 0 \implies g(y) = C

So we have

f(x,y) = xy + C \implies \nabla f = \vec F

is indeed conservative.

(b) By the gradient theorem, the integral is path-independent and

\displaystyle \int_C \vec F \cdot d\vec r = f(5,4) - f(0,2) = \boxed{20}

for <em>any</em> path C that starts at (0, 2) and ends at (5, 4).

Lisa [10]1 year ago
7 0

Answer:

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Step-by-step explanation:

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