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Jacobian matrix
The Jacobian of a vector function is a matrix of the partial derivatives of that function.
Compute the Jacobian matrix of [x*y*z, y^2, x + z] with respect to [x, y, z].
syms x y z jacobian([x*y*z, y^2, x + z], [x, y, z])
ans = [ y*z, x*z, x*y] [ 0, 2*y, 0] [ 1, 0, 1]
Now, compute the Jacobian of [x*y*z, y^2, x + z] with respect to [x; y; z].
jacobian([x*y*z, y^2, x + z], [x; y; z])
The Jacobian of a scalar function is the transpose of its gradient.
Compute the Jacobian of 2*x + 3*y + 4*z with respect to [x, y, z].
syms x y z jacobian(2*x + 3*y + 4*z, [x, y, z])
ans = [ 2, 3, 4]
Now, compute the gradient of the same expression.
gradient(2*x + 3*y + 4*z, [x, y, z])
ans = 2 3 4
The Jacobian of a function with respect to a scalar is the first derivative of that function. For a vector function, the Jacobian with respect to a scalar is a vector of the first derivatives.
Compute the Jacobian of [x^2*y, x*sin(y)] with respect to x.
syms x y jacobian([x^2*y, x*sin(y)], x)
ans = 2*x*y sin(y)
Now, compute the derivatives.
diff([x^2*y, x*sin(y)], x)
ans = [ 2*x*y, sin(y)]
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