looking for where can i learn this problems of numerical analysis (specifically problem 2)
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HallsofIvy
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For (2) what are \(\displaystyle \nabla\) and \(\displaystyle \Delta\)? How are they defined in your text book?
HallsofIvy
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Are you taking a course? Do you have a textbook? What about a teacher? "Nabla" is defined here: Nabla -- from Wolfram MathWorld .
Nabla, \(\displaystyle \nabla\), is a "differential operator" that, in an xy-coordinate system is \(\displaystyle \frac{\partial}{\partial x}+ \frac{\partial}{\partial y}\) and in an xyz-coordinate system is \(\displaystyle \frac{\partial}{\partial x}+ \frac{\partial}{\partial y}+ \frac{\partial}{\partial z}\).
For example, \(\displaystyle \nabla f(x,y,z)\) where f is a scalar valued of x, y, and z is \(\displaystyle \frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}+ \frac{\partial f}{\partial z}\) (called "grad f", "gradient of f") and is the vector that points in the direction of fastest increase of f and has length the rate of increase (derivative) in that direction.
Or, if \(\displaystyle \vec{f}= f_1(x,y,z)\vec{i}+ f_2(x,y,z)\vec{j}+ f_3(x,y,z)\vec{k}\) is a vector valued function the "dot product" of \(\displaystyle \nabla\) and \(\displaystyle \vec{f}\), \(\displaystyle \nabla\cdot \vec{f}= \frac{\partial f_1}{\partial x}+ \frac{\partial f_2}{\partial y}+ \frac{\partial f_3}{\partial z}\) (called "div f", divergence), measures how fast the vector functions spreads out. Finally, the cross product of \(\displaystyle \nabla\) and \(\displaystyle \vec{f}= \left(\frac{\partial f_2}{\partial z}- \frac{\partial f_3}{\partial f_2}{\partial z}\right)\vec{i}+ \left(\frac{\partial f_3}{\partial x}- \frac{\partial f_1}{\partial z}\right)\vec{j}+ \left(\frac{\partial f_1}{\partial y}- \frac{\partial f_2}{\partial x}\right)\vec{k}\). The "curl" measures the rotation of the vector field.
\(\displaystyle \Delta\) is much simpler. In introductory Calculus \(\displaystyle \Delta x\) is used to mean a small change in x. For example, the derivative of y(x) can be defined by \(\displaystyle \frac{dy}{dx}= \)\(\displaystyle \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}\)\(\displaystyle = \lim_{\Delta x\to 0}\frac{y(x+\Delta x)- y(x)}{\Delta x}\). That is, \(\displaystyle \Delta x\) is a small change in x and \(\displaystyle \Delta y\) is a small change in y, the change from \(\displaystyle y(x)\) to \(\displaystyle y(x+ \Delta x)\).
Nabla, \(\displaystyle \nabla\), is a "differential operator" that, in an xy-coordinate system is \(\displaystyle \frac{\partial}{\partial x}+ \frac{\partial}{\partial y}\) and in an xyz-coordinate system is \(\displaystyle \frac{\partial}{\partial x}+ \frac{\partial}{\partial y}+ \frac{\partial}{\partial z}\).
For example, \(\displaystyle \nabla f(x,y,z)\) where f is a scalar valued of x, y, and z is \(\displaystyle \frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}+ \frac{\partial f}{\partial z}\) (called "grad f", "gradient of f") and is the vector that points in the direction of fastest increase of f and has length the rate of increase (derivative) in that direction.
Or, if \(\displaystyle \vec{f}= f_1(x,y,z)\vec{i}+ f_2(x,y,z)\vec{j}+ f_3(x,y,z)\vec{k}\) is a vector valued function the "dot product" of \(\displaystyle \nabla\) and \(\displaystyle \vec{f}\), \(\displaystyle \nabla\cdot \vec{f}= \frac{\partial f_1}{\partial x}+ \frac{\partial f_2}{\partial y}+ \frac{\partial f_3}{\partial z}\) (called "div f", divergence), measures how fast the vector functions spreads out. Finally, the cross product of \(\displaystyle \nabla\) and \(\displaystyle \vec{f}= \left(\frac{\partial f_2}{\partial z}- \frac{\partial f_3}{\partial f_2}{\partial z}\right)\vec{i}+ \left(\frac{\partial f_3}{\partial x}- \frac{\partial f_1}{\partial z}\right)\vec{j}+ \left(\frac{\partial f_1}{\partial y}- \frac{\partial f_2}{\partial x}\right)\vec{k}\). The "curl" measures the rotation of the vector field.
\(\displaystyle \Delta\) is much simpler. In introductory Calculus \(\displaystyle \Delta x\) is used to mean a small change in x. For example, the derivative of y(x) can be defined by \(\displaystyle \frac{dy}{dx}= \)\(\displaystyle \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}\)\(\displaystyle = \lim_{\Delta x\to 0}\frac{y(x+\Delta x)- y(x)}{\Delta x}\). That is, \(\displaystyle \Delta x\) is a small change in x and \(\displaystyle \Delta y\) is a small change in y, the change from \(\displaystyle y(x)\) to \(\displaystyle y(x+ \Delta x)\).
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i am really bad at math (really bad might be an understatement) but i want to learn
but couldn't get the courage to learn from the beginning.. because i have start from high school again.
Are you taking a course >>> yes (BSC in CSE)
Do you have a textbook >>> yes (Numerical Analysis Richard L. Burden, J. Douglas Faires, Annette M. Burden)
What about a teacher? >>> yes but too busy.... and cant access other resources because of covid.
i took some notes from class.
i understood what my teacher taught me but when it come to solve anything i just cant.
but i feel like somethings missing that i should have known from the start.
but i don't know what.
i feel so lost and the shear amount of things i have to learn just freezes me.
if i am not climbing i am falling.
but couldn't get the courage to learn from the beginning.. because i have start from high school again.
Are you taking a course >>> yes (BSC in CSE)
Do you have a textbook >>> yes (Numerical Analysis Richard L. Burden, J. Douglas Faires, Annette M. Burden)
What about a teacher? >>> yes but too busy.... and cant access other resources because of covid.
i took some notes from class.
i understood what my teacher taught me but when it come to solve anything i just cant.
but i feel like somethings missing that i should have known from the start.
but i don't know what.
i feel so lost and the shear amount of things i have to learn just freezes me.
if i am not climbing i am falling.
