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

question 171

Essay

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 curve C. $\mathbf { F } ( x , y ) = y \mathbf { i } + x \mathbf { j };$ C is the arc of the curve $y = x ^ { 4 } - x ^ { 3 }$ from $( 1,0 )$ to $( 2,8 )$ .

---

Definitions:

Epigenetic

Pertains to the study of heritable changes in gene expression that do not involve changes to the underlying DNA sequence.

Depression's Vicious Cycle

A repetitive loop where negative thoughts exacerbate depressive symptoms, which in turn generate more negative thoughts.

Behavioral Withdrawal

The act of retreating or disengaging from social interactions and activities, often as a coping mechanism for stress or anxiety.

Social Rejection

The act of being deliberately excluded or dismissed by individuals or groups in social or professional contexts.