Calculate the Flow in the Field F Along the Path $\mathbf { F } = \mathrm { e } ^ { 2 } \mathrm { i } + \frac { 1 } { \mathrm { y } } \mathrm { j } + 2 \mathbf { k } , \mathrm { C }$

question 74

Multiple Choice

Calculate the flow in the field F along the path C.
- $\mathbf { F } = \mathrm { e } ^ { 2 } \mathrm { i } + \frac { 1 } { \mathrm { y } } \mathrm { j } + 2 \mathbf { k } , \mathrm { C }$ is the curve $\mathbf { r } ( \mathrm { t } ) = 4 \mathrm { t } ^ { 2 } \mathrm { i } + 5 \mathrm { tj } + ( - 8 - 2 \mathrm { t } ) \mathbf { k } , 1 \leq \mathrm { t } \leq 3$

• A) \(4 \left( \mathrm { e } ^ { 225 } - \mathrm { e } ^ { 25 } \right) + \ln 3 - 8\)
• B) \(\frac { 8 } { 25 } \left( \mathrm { e } ^ { 225 } - \mathrm { e } ^ { 25 } \right) + \ln 3 - 8\)
• C) \(\frac { 4 } { 25 } \left( e ^ { 225 } - e ^ { 25 } \right) + \ln 3 - 8\)
• D) \(\frac { 4 } { 25 } \left( \mathrm { e } ^ { 225 } - \mathrm { e } ^ { 25 } \right) + \ln 3 + 8\)

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Definitions:

Double Chains

Structures consisting of two parallel chains of atoms or molecules bonded together, often found in silicate minerals.

Silicon Tetrahedron

A structural unit in silicate minerals, consisting of a central silicon atom covalently bonded to four oxygen atoms at the corners of a tetrahedron.

Silicate Minerals

Minerals that contain silicon and oxygen, and possibly other elements, constituting the largest family of minerals.

Silicon Tetrahedron

A fundamental building block in silicate minerals, consisting of a silicon atom surrounded by four oxygen atoms in a tetrahedral arrangement.