a.) Find a function f such that $$F=\nabla f$$ b.) use a.) to evaluate $$\int_c F\bullet dr$$ along the given curve C $$F(x,y)=xy^2\hat{i}+x^2y\hat{j}$$ $$C:\hat{r}(t)=,$$ $$0\le t \le 1$$ $$\frac{\partial}{\partial{y}}|xy^2|=x2y$$ $$\frac{\partial}{\partial{x}}|x^2y|=2xy$$ $$f_x=xy^2$$ $$f=\frac12x^2y^2+P(y)$$ $$f_y=x^2y$$ $$f=\frac12x^2y^2+Q(x)$$ $$\frac12x^2y^2+P(y)=\frac12x^2y^2+Q(x)$$ $$P(y)-Q(x)=0$$ $$f(x,y)=\frac12x^2y^2$$

Question

a.) Find a function f such that $$F=\nabla f$$
b.) use a.) to evaluate $$\int_c F\bullet dr$$ along the given curve C
$$F(x,y)=xy^2\hat{i}+x^2y\hat{j}$$
$$C:\hat{r}(t)=<t+sin\frac12\pi t, t+cos\frac12\pi t>,$$
$$0\le t \le 1$$
$$\frac{\partial}{\partial{y}}|xy^2|=x2y$$
$$\frac{\partial}{\partial{x}}|x^2y|=2xy$$
$$f_x=xy^2$$
$$f=\frac12x^2y^2+P(y)$$
$$f_y=x^2y$$
$$f=\frac12x^2y^2+Q(x)$$
$$\frac12x^2y^2+P(y)=\frac12x^2y^2+Q(x)$$
$$P(y)-Q(x)=0$$
$$f(x,y)=\frac12x^2y^2$$

anonymous · Answer

I guess I figured out part (a.)

experimentx · Answer

put all those into t's|dw:1343196362628:dw|
and then take dot product and evaluate