Let 𝐅 be a differentiable vector field and let g(x, y, z) be a differentiable scalar function. V

Let 𝐅 be a differentiable vector field and let g(x, y, z) be...

Key Concepts

Divergence

Divergence is a scalar measure of a vector field’s tendency to originate from or converge into a point. In the context of vector calculus, it is given by the dot product of the del operator with the vector field. When the divergence is applied to a product of a scalar function and a vector field, the product rule helps to distribute the differentiation, leading to one term involving the divergence of the vector field and another term involving the gradient of the scalar function dotted with the vector field.

Curl

Curl is a vector operator that measures the rotation of a vector field around a point. It is defined as the cross product of the del operator with the vector field. Applying the curl to a product of a scalar function and a vector field also requires a product rule, resulting in one term that multiplies the scalar function with the curl of the vector field and a second term that involves the cross product of the gradient of the scalar function with the vector field.

Gradient

The gradient of a scalar function is a vector that points in the direction of the greatest rate of increase of the function, with its magnitude being the rate of increase. It is instrumental in formulating the product rules for divergence and curl when a scalar function is multiplied with a vector field, allowing these derivative operations to be decomposed into simpler components.

Product Rule in Vector Calculus

The product rule in vector calculus generalizes the familiar product rule from single-variable calculus to operations involving scalar and vector fields. When a scalar function multiplies a vector field, the gradient, divergence, and curl of the product are expressed as sums of terms involving derivatives of both functions. This rule is essential for understanding how differential operators act on combinations of functions.

Transcript

00:01 So we have a g differential function of x y z and we have a f differential differential field i would like to check to check the following if you would like to check data the divergence of g times f is equal to g times g times the divergence of f plus the divergence of the divergence of well the grad of g dot product of f so the way we can do this is to just compute so so g is a function of x, y, z, f depends on x, y, c, so the divergence of these would be equal to the derivative of all these.

01:44 G times f defines a new vector field that is in components is going to be g f1, g, f2, and g f3 if the vector field f is written as f1 in the i direction plus if 2 in the j direction plus f3 long k so that well this would be equal to differentiating this vector field which is that so relative with respect to x of g f1 plus negative with respect to y of g of f2 plus negative with respect to z of g of f3 so well by the protocol here you have that a negative of g if one would be that if partial with respect to x x of g times f1 plus g times partial with respect to x of f1 differentiating this with respect to y would be partial of g with respect partial of g with respect to y times f2 plus g times partial of y partial with respect to y of f2 and lastly partial of z with respect of g with respect to z times f three plus g times partial of c of f3 with respect to c so now well um look at these terms so these terms can be seen as uh oh they are just those terms so let's separate these terms the green ones and the blue ones.

04:24 So if you write just the green ones, so this is right of, well this is partial with respect to x of g times f1 plus partial with respect to y of g times of two plus partial with respect to c of g times of three.

04:54 And here we have g, let's factor the g out of each of these terms, g terms partial of 1 with respect to x, plus partial of f2 with respect to y, plus partial of f3 with respect to z.

05:16 So well now you can see that this term can be seen as a dot part.

05:24 That part with the vector, partial of g with respect to x in the first component, partial of y with respect to g, partial of g with respect to y, partial of g with respect to y, partial of g with respect to c in the third component...

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