# Help multivariable calculus

1. Dec 16, 2008

### Carl140

1. The problem statement, all variables and given/known data

Let f: R^n -> R be defined as follows:

f(x) = x*L(x) where * denotes the standard inner product and L: R^n -> R^n is a linear
function.
I'm trying to find the directional derivative f'(x;u).

2. Relevant equations

I know that f'(x;u) (the directional derivative of f(x) in the direction of the unit vector u)
is equal to gradient(f(x)) * u where * denotes inner product.

3. The attempt at a solution

(Not sure if this step is correct).
Then because L: R^n -> R^n is a linear map the differential of L is L again, no?

Conclusion: the directional derivative in the direction of u is then (2L(x)*gradient(x))*u where u is a unit vector.

Is this correct? I'm a little bit confused about the part of L being a linear map, is it correct
to state that the gradient is again L itself?

2. Dec 16, 2008

### Dick

You are writing expressions that don't really make sense. Think about what f(x) looks like in components. It's f(x)=x_i*L_ij*x_j (where x_i are the components of x, L_ij is the matrix representing L and i and j are summed from 1...n). The kth component of the gradient is df/x_k. Can you think how to write that concisely using matrices?

3. Dec 16, 2008

### Carl140

I think f(x) = x^t * A * x where x represents the column vector and A is the matrix
which represents L. Is this correct so far?