Find a Function $f$ Such That $\mathbf { F } = \nabla f$

question 104

Short Answer

Find a function $f$ such that $\mathbf { F } = \nabla f$ , and use it to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the given curve $C$ .
$$\begin{array} { l } \mathbf { F } ( x , y ) = x ^ { 3 } y ^ { 4 } \mathbf { i } + y ^ { 3 } x ^ { 4 } \mathbf { j } \\C : \mathbf { r } ( t ) = \sqrt { t } \mathbf { i } + \left( 1 + t ^ { 3 } \right) \mathbf { j } , 0 \leq t \leq 1\end{array}$$

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Definitions:

Foreground-background

A concept in visual perception where the focus is on distinguishing the main subject (foreground) from the surrounding environment or background.

Perceptual Apathy

A lack of interest or concern regarding perception, leading to indifference to sensory experiences or stimuli.

Figure-ground

is a visual perception principle that involves the segmentation of visual information into "figure" (the object of focus) and "ground" (the background or surroundings).

Foreground-background

The distinction between the central object of visual attention (foreground) and everything else that forms the context or backdrop (background).