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mLine 54:
Line 54: − − We've discussed partial derivatives, but it seems there should be a way to find a derivative which is in some sense "complete." We'll address this concept for vector fields in the next lecture, but for now let's examine the gradient, a differential operator for scalar functions. For every point in a scalar field, the '''gradient''' is a vector in the direction of greatest change, with length equal to the rate of change in that direction (ie, the directional derivative). − − The gradient is easily computed: − − <math>\vec{grad} f = \vec{i}\partial_x f+ \vec{j}\partial_y f + \vec{k}\partial_z f</math>. − − If we wanted to construct a more concise version of this, we can always form the "vector" often called '''del''' or '''nabla''': − − <math>\nabla = \vec{i}\frac{\partial}{\partial x} + \vec{j}\frac{\partial}{\partial y} + \vec{k}\frac{\partial}{\partial z}</math>. − − Then, if we just take this operator of ''f'', we get the gradient: <math>\vec{grad} f = \nabla f</math>. Remember this operator; we'll encounter it again.
→Lecture 3: hehe i got confused
===[[Calc3.3|Lecture 3]]===
===[[Calc3.3|Lecture 3]]===
*'''The Gradient'''
*'''The Gradient'''
*'''Directional Derivatives'''
*'''Directional Derivatives'''