This work represents our effort to present the basic concepts of vector and tensor analysis. However, the direction of \(-c \overrightarrow{\mathbf{A}}\) is opposite of \(\overrightarrow{\mathbf{A}}\) (Figure 3.6). Vector quantities also satisfy two distinct operations, vector addition and multiplication of a vector by a scalar. The Divergence Theorem relates an integral over a volume to an integral over the surface bounding that volume. (\textbf{C} \times \textbf{A}) = \textbf{C} . We can represent vectors as geometric objects using arrows. (\frac{A}{g}) = \frac{g(\nabla . There are two defining operations for vectors: Vectors can be added. Vector addition satisfies the following four properties: The order of adding vectors does not matter; \[\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}} = \overrightarrow{\mathbf{B}}+\overrightarrow{\mathbf{A}}\]Our geometric definition for vector addition satisfies the commutative property (3.1.1). Notes of the vector analysis are given on this page. (\textbf{B} \times \textbf{C})$, Geometrically, $\left|\textbf{A} . Multiplying vectors by scalars is very useful in physics. For every vector \(\overrightarrow{\mathbf{A}}\) there is a unique inverse vector \(-\overrightarrow{\mathbf{A}}\) such that \[\overrightarrow{\mathbf{A}} + (-\overrightarrow{\mathbf{A}}) = \overrightarrow{\mathbf{0}}\] The vector \(-\overrightarrow{\mathbf{A}}\) has the same magnitude as \(\overrightarrow{\mathbf{A}}\), \(|\overrightarrow{\mathbf{A}}|=|-\overrightarrow{\mathbf{A}}|=A\) but they point in opposite directions (Figure 3.5). We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Let \(c\) be a real number. v = \frac{\partial v_{x}}{\partial x} + \frac{\partial v_{y}}{\partial y} + \frac{\partial v_{z}}{\partial z}$, $\nabla \times F = (\frac{\partial F_{z}}{\partial y} – \frac{\partial F_{y}}{\partial z}) \textbf{i} + (\frac{\partial F_{x}}{\partial z} – \frac{\partial F_{z}}{\partial x}) \textbf{j} + (\frac{\partial F_{y}}{\partial x} – \frac{\partial F_{x}}{\partial y}) \textbf{j}$, Vector derivatives in cylindrical and spherical coordinates. Volume I begins with a brief discussion of algebraic structures followed by a rather detailed discussion of the algebra of vectors and tensors. We can add two forces together and the sum of the forces must satisfy the rule for vector addition. Three numbers are needed to represent the magnitude and direction of a vector quantity in a three dimensional space. There is a unique vector, \(\overrightarrow{\mathbf{0}}\), that acts as an identity element for vector addition. Voltage, current, time, and 1D position will continue to be quantities of inter-est, but more is needed to prepare for future chapters. \textbf{C})$. $\nabla . Let \(b\) and \(c\) be real numbers.Then\[(b+c) \overrightarrow{\mathbf{A}}=b \overrightarrow{\mathbf{A}}+c \overrightarrow{\mathbf{A}}\]Our geometric definition of vector addition and scalar multiplication satisfies this condition as seen in Figure 3.8. Vector analysis: A text-book for the use of students of mathematics and physics… \textbf{C}) + \textbf{B} (\textbf{A} . Vector quantities also satisfy two distinct operations, vector addition and multiplication of a vector by a scalar. Notify me of follow-up comments by email. Then, \[c(\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}})=c \overrightarrow{\mathbf{A}}+c \overrightarrow{\mathbf{B}}\]. The number 1 acts as an identity element for multiplication, \[1\overrightarrow{\mathbf{A}} = \overrightarrow{\mathbf{A}}\], Dividing a vector by its magnitude results in a vector of unit length which we denote with a caret symbol, \[\hat{\mathbf{A}}=\frac{\overrightarrow{\mathbf{A}}}{|\overrightarrow{\mathbf{A}}|}\], Note that \(|\hat{\mathbf{A}}|=|\overrightarrow{\mathbf{A}}| /|\overrightarrow{\mathbf{A}}|=1\). 2 Chapter 1 Vector Analysis B C A Figure 1.1 Triangle Law of Vector Addition B A C F E D Figure 1.2 Vector Addition Is Associative this representation, vector addition C = A +B (1.1) consists of placing the rear end of vector B at the point of vector A (head to tail rule). B} = \left|A\right|\left|B\right|cos \theta$, Distributive: $\textbf{A . For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. - Direction cosine of a vector. (\nabla g)}{g^{2}}$, $\nabla \times (\frac{A}{g}) = \frac{g (\nabla \times A) + A \times (\nabla g)}{g^{2}}$, $\nabla ^{2} T = \nabla . Place the tail of the arrow that represents \(\overrightarrow{\mathbf{B}}\) at the tip of the arrow for \(\overrightarrow{\mathbf{A}}\) as shown in Figure 3.2a. Let \(b\) and \(c\) be real numbers. Vector physicsis the study of the various forces that act to change the direction and speed of a body in motion. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. The length of the arrow corresponds to the magnitude of the vector. The gradient $\nabla T$ points in direction of maximum increase of the function T. The magnitude $\left| \nabla T \right|$ gives the slope (rate of increase) along this maximal direction. (\nabla \times A) – A . A) – A . A)$, $\nabla (\frac{f}{g}) = \frac{ g\nabla f – f \nabla g}{g^{2}}$, $\nabla . We can understand this geometrically because in the head to tail representation for the addition of vectors, it doesn’t matter which vector you begin with, the sum is the same vector, as seen in Figure 3.3. The simplest prototype vector is given by the dis­ (fA) = f (\nabla . B) = A \times (\nabla \times B) + B \times (\nabla \times A) + (A . If you spot any errors or want to suggest improvements, please contact us. We can multiply a force by a scalar thus increasing or decreasing its strength. We can multiply a force by a scalar thus increasing or decreasing its strength. Vector analysis: A text-book for the use of students of mathematics and physics, (Yale bicentennial publications) [J. Willard Gibbs, Edwin Wilson] on Amazon.com. Have questions or comments? A) + (\nabla . We can multiply a force by a scalar thus increasing or decreasing its strength. A vector is a quantity that has both direction and magnitude. Let a vector be denoted by the symbol \(\overrightarrow{\mathbf{A}}\). 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