Solve the Problem $\mathrm { M } = \frac { \mathrm { y } } { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } }$

question 20

Essay

Solve the problem.
-Let $\mathrm { M } = \frac { \mathrm { y } } { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } }$ and $\mathrm { N } = \frac { - \mathrm { x } } { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } }$ . Show that $\int _ { \mathrm { C } } \mathrm { Mdx } + \mathrm { Ndy } z \int _ { \mathrm { R } } \left( \frac { \partial \mathrm { N } } { \partial \mathrm { x } } - \frac { \partial \mathrm { M } } { \partial \mathrm { y } } \right) \mathrm { dx } \mathrm { dy }$ , where $\mathrm { R }$ is the region bounded by the unit circle $C$ centered at the origin. Why is Green's Theorem failing in this case?

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Definitions:

Estimated Residual Value

The projected value of an asset at the end of its useful life, used in calculating depreciation.

Straight-Line Basis

An approach to determine depreciation or amortization by equally distributing an asset's cost throughout its usable life.

Residual Value

The estimated value of an asset at the end of its useful life, often considered in depreciation calculations.