Finding the Gradient of an Integral?

Homework Statement

https://dl.dropbox.com/u/64325990/Capture.PNG [Broken]

I'm not even sure where to start :O

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$$f(0+\Delta x,0+\Delta y)\approx f(0,0)+\frac{\partial f}{\partial x}(0,0)\Delta x+\frac{\partial f}{\partial y}(0,0)\Delta y$$

This uses the gradient in the sense of

$$df\approx \nabla f\cdot(dx,dy)$$

Though I'm sorting of putting together whatever notation comes to mind, so let us know if you have notes close to this but aren't sure how they relate, or any other questions.

Curious3141
Homework Helper

Homework Statement

https://dl.dropbox.com/u/64325990/Capture.PNG [Broken]

I'm not even sure where to start :O

Start with the Fundamental Theorem of Calculus to work out $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial x}$.

For the latter, you might find it more helpful to switch the bounds and put a negative sign on the integral.

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SammyS
Staff Emeritus
Homework Helper
Gold Member

Homework Statement

https://dl.dropbox.com/u/64325990/Capture.PNG [Broken]

I'm not even sure where to start :O
You have received two very good tips.

I'll make my response more concrete.

cos(t2) is integrable, but not in closed form with elementary functions.

Let G(t) be an anti-derivative of cos(t2).

Then of course, cos(t2) is the derivative of G(t).

Using this to evaluate your integral gives:

$\displaystyle \int_{x}^{y}{\cos(t^2)}\,dt=G(y)-G(x)\ .$

Now take the gradient of that.

Can you take it from here?

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