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Vector Calculus

Normal gradient vector

Use the normal gradient vector to write an equation of the line (or plane) tangent to the given curve (or surface) at the given point P: x^(1/3) + y^(1/3) + z^(1/3) = 1; P(1, -1, 1)

Gradient vector

Find the gradient vector f at the indicated point P: f(x, y, z)=(x^2 + y^2 + z^2) ; P(17, 3, 2) : is the square root of

Vectors : Dot Product

Let u=<-2,1> v=<3,4> w=<-5,12> Use properties of the dot product. u(Dot)(v-w) or u.(v-w)

Vectors : Planes, Points, Cross Product and Dot Product

(1) a. Find the (vector) equation of the plane passing through the points (1,2,-2), (-1,1,-9), (2,-2,-12). b. Find the (vector) equation of the plane containing (1,2,-1) and perpendicular to (3,-1,2). (2) Suppose a, b, c are non zero vectors. a. Explain why (a x b) x (a x c)

Linear Alegbra : Vectors

Find equations for the indictated geometrical objects The line through the point P=(1,1,1) and perpendicualar to the plane 4x-2y+6z=3

Linear alegbra

Consider the following elements of the vector space P3 of all polynomials of degree less than or equal to 3. p(x)= x-1, q(x)=x+x2, r(x)= 1+x2-x3 Do these three polynomials form a basis for P3?

Vector analysis

Apply Stroke's theorem to evaluate the integral over C of (ydx + zdy + xdz), where C is the curve of intersection of the unit sphere x^2+y^2+z^2=1 and the plane x+y+z=0, traced anticlockwise viewed from the side of the positive x-axis

Vector Calculus

F(x,y) = x^3 + y - xy + 1 a) Are there points on the curve y = (x - 1)^2 where Gradient f is perpendicular to the curve? b) Find the absolute maximum and minimum of the function in the region 1 >= x >= 0 and y >= 0.


Let G be k-critical graph with respect to vertex-arboricity (k>=3). Prove that for each vertex v of G, the graph G-v is not (k-1)-critical with respect to vertex-arboricity.

Vector Subspace of the Vector Space

1. Determine whether the following sets W are vector subspaces of the vector space V. a. V=R^4, A and B are two 3 X 4 matrices. W={X is an element of R^4:Ax-3Bx=0}. b. V=C', W={f is an element of C': f(x+3)=f(x)+5 c. V=P, W={f is an element of P: f'(2)=0} d. V=C', W={f is an element of C': The integral of f(x)dx fr