Vector Revision
What is a vector, and what does the dot product represent?
A quantity that has both magnitude and direction. The dot product is the directional multiplication, that is, a measure of how closely two vectors align, in terms of the directions they point.
The partial derivative of 4xy3 with respect to y is...
12xy2
If f(x,y) = 2x3 + y3 then the gradient ∇ f(x,y) is...
6x2 i + 3y2 j
If a is a vector represented by 2i + 6j and b is also a vector, represented by -4i + 8j, a.b is...
40
The partial derivative of 8x2yz3 with respect to z is...
24x2yz2
Consider f(x, y) = 7xy - x3y + x2. The gradient vector is...
(7y - 3x2y + 2x) i + (7x - x3) j
If vector a = 4i + 2j - k and vector b = -3i + 6j, what is 2a + b?
5i + 10j - 2k
If f(x,y) = 6x2y + 5xy2, what are the partial derivatives with respect to x and y?
WRT x = 12xy + 5y2
WRT y = 6x2 + 10xy
If f(x,y) = sin(x3y), then the gradient ∇ f(x,y) is...
[cos (x3y) * 3x2y] i + [cos (x3y) * x3] j
If a is a vector represented by 6i + 2j - 3k and b is also a vector, represented by -4i + j - 3k, a x b is...
-3i + 30j + 14k
If f(x,y,z,p) = 6x2y4z3p3, the partial derivatives with respect to x,y,z and p are...
WRT x = 12xy4z3p3
WRT y = 24x2y3z3p3
WRT z = 18x2y4z2p3
WRT p = 18x2y4z3p2
If f(x,y,z) = 3ex + z2 - 6xy + 2y6, then the gradient vector is...
(3ex - 6y) i + (-6x + 12y5) j + (2z) k