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The solid upper hemisphere of the sphere of radius 2 centered at the origin

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Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 1

Three-Dimensional Coordinate Systems

Vectors

Missouri State University

Harvey Mudd College

University of Nottingham

Boston College

Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

03:42

Find the radius and center…

02:55

Find the center and radius…

02:09

02:11

03:52

01:23

Find the sphere's cen…

Yeah, let's write mathematical statements that define the region of the upper hemisphere of the solid sphere centered at the origin with radius too. So the silence here, Senator, the origin with radius too. It's gonna be a sphere right here. The boundary of that sphere. It's gonna have the equation X square plus why square plus the square equals two square. But we don't just want the edge of this year. We want this solid sphere. Yes, we want that and all the points on the inside X Where? Let's slice where, plus the square, less than or equal 24 But we only want the upper half. So you want to cut it off right here along this circle and not include the bottom half? Well, that's an extra restriction just on Z. We're forcing Z to be non negative and so we still want all the points inside the sphere with the added a requirement that Z can't be negatives and that will give us region that we want

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