Partial derivative with respect to a vector

In summary, Partial derivative notation for taking the partial derivative of a function f with respect to a vector x is also referred to as a gradient and is known as matrix calculus notation. This notation allows for the representation of a function as a linear function plus a non-linear function, with the linear function being the derivative of f at a given point. This notation is useful for finding tangent line approximations and for representing linear transforms.
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TheOldHag
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I've come across using partial derivative notation for taking the partial derivative of a function f with respect to a vector x. I've never seen this before. It is also being referred to as a gradient. However, I have only seen gradients where all variables in the space are featured in the result vector. In this case, the result is a vector but not with components representing each dimension in the space. On wikipedia I've seen this referred to as matrix calculus notation. I would like to know a bit more about this in broad terms. For instance, for a space x1, x2, x3, x4 if I take the partial derivative with respect to a vector x1,x2 is that result vector valued function pointing in the direction of steepest ascent similar to a gradient but only for x1 and x2? Any other pointers appreciated.
 
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If f is a function from [itex]R^n[/itex] to [itex]R^m[/itex] then we can always write it, around x= p, as a linear function plus a non-linear function: [itex]f(x)= f(p)+ L(x- p)+ N(x)[/itex] where L is linear and N is non-linear. Of course, if f is continuous at x= p, we must have [itex]\lim_{x\to p} N(x)= 0[/itex]. In addition, we say that f is "differentiable at x= p" if and only N(p) goes to zero "faster than linearly"- specifically that [itex]\lim_{x\to p}N(x)/|x-p|= 0[/itex].

In that case we say that L(p) is the "derivative of f(x) at x= p". If, for example, m= n= 1, the usual "real valued function of a single variable", [itex]f(x)= f'(p)(x- p)[/itex] is the tangent line approximation to f- and "L" is f'(p). In the case that n= 1 and m> 1, we can think of L as being a vector which, multiplied by x- p, gives the vector value approximating f(p). That is, L is the vector of derivatives of the coordinate functions. If f(x) is a real valued function of several variables: n> 1 and m=1, L is the vector whose dot product with the variable gives f(x). That would be [itex]\nabla f[/itex].

More generally, If f is from [itex]R^n[/itex] to [itex]R^m[/itex], L is a linear transform that maps the n-vector x- p to an m-vector: it can be represented (in a given coordinate system) by a matrix with n columns and m rows.
 
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What is a partial derivative with respect to a vector?

A partial derivative with respect to a vector is a mathematical calculation that measures the rate of change of a function with respect to one specific variable in a multi-variable system. It involves taking the derivative of the function with respect to only one variable while holding all other variables constant.

How do you calculate a partial derivative with respect to a vector?

To calculate a partial derivative with respect to a vector, you first need to determine which variable you are taking the derivative with respect to. Then, you use the standard rules of differentiation to calculate the derivative of the function with respect to that variable while treating all other variables as constants. This can be done using the chain rule or product rule, depending on the complexity of the function.

Why is it important to calculate partial derivatives with respect to a vector?

Partial derivatives with respect to a vector are important because they allow us to understand the sensitivity of a function to changes in specific variables. This is especially useful in fields such as physics and economics, where multiple variables are involved and the behavior of a system can be better understood by analyzing the individual components.

What is the difference between a partial derivative and a total derivative with respect to a vector?

The main difference between a partial derivative and a total derivative with respect to a vector is the number of variables involved. A partial derivative only considers the change in one specific variable, while holding all other variables constant. A total derivative, on the other hand, takes into account the change in all variables simultaneously, including those that are dependent on the variable being differentiated.

Can a partial derivative with respect to a vector be negative?

Yes, a partial derivative with respect to a vector can be negative. This indicates that the function is decreasing in value with respect to that specific variable. It is also possible for a partial derivative to be positive or zero, depending on the behavior of the function and the direction of change in the vector.

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